Understanding Factor Theorem Solutions

In this video I'm going to go through the solutions to the questions on the factor theorem. If you wish to try the questions there's a link in this video's description. For this question we need to show that x plus 4 is a factor. To do that we'll substitute in negative 4. So f of negative 4 is negative 4 cubed plus 5 lots of negative 4 squared plus 2 lots of negative 4 take away 8. Since it's a show that we need to do all of the steps. So negative 4 cubed is negative 64, negative 4 squared is 16, and then times that by 5 gives you positive 80, 2 times negative 4 is negative 8, and then negative 8. And if you do this you get 0, which shows that x plus 4 is a factor. For part b we need to factorise f of x, so what we'll do is we'll take the polynomial and then we'll divide it by the factor x plus 4. So x cubed divided by x is x squared, then multiply this back, x squared times x is x cubed, and then x squared times 4 is plus 4x squared. Now we subtract these, the x cubes will cancel, 5x squared take 4x squared is just 1x squared. Bring down the next term, and then we'll do x squared divided by x, which is just x, multiply that back, x times x is x squared and x times 4 is plus 4x. Then subtract these, the x squared will cancel, and 2x take 4x is negative 2x. Bring down the final term and then do negative 2x divided by x which is negative 2 and then multiply this back through gives you negative 2x subtract 8. Subtract these and you get 0. So we found that this could be expressed as x plus 4 times by a quadratic x squared plus x take away 2. But this quadratic can also factorize to x plus 2 x minus 1. For this question we need to show that 2x minus 1 is a factor. To do this we'll do f of 1 half. So we need to do 2 lots of 1 half cubed, plus 13 lots of 1 half squared, plus 13 lots of 1 half, take away 10. Now 1 half cubed is 1 eighth, so we've got 2 lots of 1 eighth. 1 half squared is a quarter, so we've got 13 lots of a quarter, and then 13 lots of a half, and then take away 10. So 2 times 1 eighth is 2 eighths, which would simplify to 1 quarter. Then we've got 13 quarters. 13 halves and take away 10. Now it would make sense to write all of these over 4, so the 13 halves will become 26 over 4 and the negative 10 will become negative 40 over 4. Now it should become clear that this equals 0, so we've finished. For part b we need to factorise, so what we'll do is take the polynomial and divide it by the factor. So 2x cubed divided by 2x is x squared, multiply this back through, We get 2x cubed, take away x squared. Now we subtract these, the 2x cubed will cancel, and 13x squared take away 1x squared is 14x squared. Now bring down the next term, 14x squared divided by 2x gives you 7x, and then multiply this back through, 7x times 2x is 14x squared, and 7x times negative 1 is negative 7x. Now subtract these, the 14x squareds will cancel, and 13x take away negative 7x is 20x. Then bring down the final term, 20x divided by 2x is plus 10, and multiply this back through, gives you 20x take away 10, which when you subtract these two, gives you 0. So we've found that this is the linear part times the quadratic part, and this quadratic part can further factorise to x plus 5 x plus 2. For this one we need to show that x plus 2 is a factor, so we'll do f of negative 2, which is negative 2 cubed, take away 5 lots of negative 2 squared, take away 2 lots of negative 2, add 24. Negative 2 cubed is negative 8, negative 2 squared is 4, and then times negative 5 is negative 20, negative 2 times negative 2 is positive 4, and then plus 24. And if you do this you get 0, so x plus 2 must be a factor. For part b of this question we need to solve f of x equals zero. To do this we need to fully factorize f of x. Now we have one of the factors so we'll just take f of x and divide it by that factor. So we do x cubed divided by x which is x squared, then multiply the x squared back through, x squared times x is x cubed and x squared times 2 is plus 2x squared. Then we subtract these, x cubed take x cubed to zero, Minus 5x squared, take away 2x squared, is a negative 7x squared. Then bring down the next term and we'll do negative 7x squared divided by x, which is negative 7x, and multiply this back through. Negative 7x times x is negative 7x squared, and negative 7x times positive 2 is negative 14x. Now subtract these, negative 7x squared will cancel, and negative 2x subtract negative 14x is 12x. Then bring down the final term, Do 12x divided by x which is plus 12 and then multiply this back through, we'll get you 12x plus 24 and you subtract these and you'll get 0. So we can now factorise this, we have x plus 2 times by the quadratic and remember it equals 0 because we're solving the equation now. We can factorise this further, the quadratic will factorise to x minus 3 x minus 4 and now we can see our solutions. So the first bracket gives us x equals negative 2 the second bracket x equals 3, and the final bracket x equals 4. For part a of this question we need to show that 4x plus 1 is a factor. To do this we'll do f of negative a quarter. So that'll be 4 lots of negative a quarter cubed, take away 11 lots of negative a quarter squared, plus 5 lots of negative a quarter, plus 2. Now negative a quarter cubed is negative 1 over 64, so we've got 4 lots of negative 1 over 64, Negative a quarter squared is positive 1 over 16, so it's take away 11 lots of 1 16th, and then we've got plus 5 lots of negative a quarter plus 2. If we do 4 lots of negative 1 over 64, we get negative 4 over 64, which simplifies to negative a 16th, and then we do negative 11 over 16, take away 5 over 4, plus 2. Now we need all of these to be over 16, so the last two terms will change. Negative 5 over 4. would become negative 20 over 16 and the plus 2 will become 32 over 16. Now it's clear that this equals 0 therefore 4x plus 1 is a factor. For part b we need to solve f of x equals 0. To do this we need to fully factorize it. So we'll take the polynomial and divide it by the factor that we have. 4x cubed divided by 4x is x squared. Multiply this x squared back through you get 4x cubed plus x squared. Then we make a subtraction 4x cubes will cancel, negative 11x squared take away x squared is negative 12x squared. Bring down the next term, and then do negative 12x squared divided by 4x, which gives you negative 3x, then multiply this back through, and you'll get negative 12x squared take away 3x, and then do a subtraction, the negative 12x squareds will cancel, and then 5x take away negative 3x gives you 8x. Bring down the final term, 8x divided by 4x is just plus 2, and then multiply this back through and you'll get 8x plus 2 which when you subtract gives you 0. So we've now been able to factorize this so we could write it as 4x plus 1 times y our quadratic equals 0. The quadratic will factorize further so that could be written as x take away 1 x take away 2 and then we get our solutions. The first bracket will give us x equals negative a quarter, the second bracket gives us x equals 1 and the final bracket gives us x equals 2. For this question we have to share x minus 2 as a factor so we'll do f of 2. So that's 3 lots of 2 cubed, take away 10 lots of 2 squared, plus 4 lots of 2, plus 8. 2 cubed is 8, so 3 eighths are 24, 2 squared is 4, and negative 10 times 4 is negative 40, 4 twos are 8, and then plus 8. This gives you 0, so we've shown it's a factor. For part b we need to solve f of x equals 0. To do this we need to factorise it fully, so again we'll take our polynomial and divide it by the linear factor. 3x cubed divided by x is 3x squared. Multiply this back through and you get 3x cubed, take away 6x squared. Now we do a subtraction, the 3x cubed will cancel, and negative 10x squared take away negative 6x squared gives you negative 4x squared. Bring down the next term, negative 4x squared divided by x gives you negative 4x, multiply this back through, negative 4x squared plus 8x. Now we subtract, the negative 4x squared will cancel, so 4x take 8x is negative 4x, bring down this final term and then do negative 4x divided by x, which gets you negative 4, and multiply that negative 4 back through, and you'll get negative 4x plus 8, take away these and you get 0. So we can write f of x as the linear part, x minus 2, times this quadratic part. which equals 0. This quadratic part can also factorise further, so it would be 3x plus 2, x take away 2. So for this one the first bracket gives you x equals 2, the second bracket gives you x equals negative 2 thirds, and the final bracket gives you a repeated solution x equals 2 again so we don't need to write that down. In this question we're told that x plus 3 is a factor, this means that if we do f of negative 3 we know we get 0. So let's substitute that in, so negative 3 cubed plus a lots of negative 3 squared take away 21 lots of negative 3 take 18. Now we know this must equal 0, so if we simplify the left hand side, negative 3 cubed is negative 27, negative 3 squared is positive 9 and then times a gives us plus 9a, negative 21 times negative 3 is plus 63 and then take 18 equals 0. If we collect up these constant terms we end up with 9a and then plus 18 equals 0. If we rearrange this we get 9a equals negative 18. and then divide both sides by 9, we'll get a equals negative 2. And now to factorise this we'll do long division, so we'll take the polynomial, remember we now know the value of a that was negative 2, so it's negative 2 x squared, and we divide this by the linear factor. So x cubed divided by x is x squared, multiply this back through and you'll get x cubed plus 3 x squared, if you subtract these the x cubes cancel, Negative 2x squared take away 3x squared is negative 5x squared. Then we bring down the next term and negative 5x squared divided by x gives you negative 5x. And then we multiply this back through. Negative 5x squared take away 15x. Now we subtract these, the negative 5x squareds will cancel. Negative 21x take away negative 15x gives you negative 6x. And bring down the final term. Negative 6x divided by x gives you negative 6. and then multiply it back through, then you get negative 6x take 18, which of course when you subtract gives you 0. So we've got x plus 3 times by the quadratic factor, and then we can factorise this quadratic one into x minus 6 x plus 1. Here we have the equation of a graph along with the sketch of it. We're asked to find the value of q. Now if you substitute x equals 0 into this, All of the terms with x will disappear so we're just left with y equals q. Now when x equals 0 we're on the y-axis and this graph crosses the y-axis at negative 20, therefore q must be negative 20. So we have a graph of this equation and notice I've updated q to be negative 20 and we need to find the value of p. Notice the graph crosses the x-axis at the number 5 here. This means if you substitute 5 in you'll get 0 out. And if this is the case by the factor theorem, x minus 5 must be a factor. So we know that f of 5 must equal 0. So if we substitute 5 in, we get 5 cubed minus 5 squared plus p lots of 5 take 20 equals 0. We can simplify this a bit, so 5 cubed is 125, 5 squared is 25 so take 25, and then plus 5p minus 20 equals 0. If we collect up the constant terms we end up with 5p plus 80 equals 0. Take 80 from both sides and you'll get 5p equals negative 80 and then divide by 5 and you'll find that p is negative 16. We now know the values of p and q so we know the full equation of the graph. We also know one of the factors x minus 5. So if we take our graph and divide it by x minus 5 we'll find out the other factors. So x cubed divided by x is x squared, multiply this back through and you get x cubed minus 5x squared. Do a subtraction, the x cubes cancel, negative x squared take away negative 5x squared gives you 4x squared. Bring down the next term and do 4x squared divided by x which gives you 4x. Now multiply this back through, 4x times x minus 5 gives you 4x squared take away 20x. Now subtract, the 4x squareds will cancel. And negative 16x take away negative 20x, gives you 4x, and bring down the next term. Now 4x divided by x gives you plus 4, and multiply this 4 back through, gives you 4x take 20, and if you subtract these you get 0. So we've shown that this graph could be written in the form x minus 5, x squared plus 4x plus 4. But this can also factorise into x plus 2, x plus 2. Notice we get a repeated root here, with x plus 2 coming up. up twice. If we were to solve the equation f of x equals zero, we would get the solutions 5, which we know, and negative 2. This means the point C here is at. In the question we were only asked for the x coordinate of the point C, so it's. In this question we're told that x-2 and x plus 6 are both factors, in which case if we do f, we should get 0. So if we do f, that's 2 lots of 2 cubed, plus 11 lots of 2 squared, plus a lots of 2, plus b, and this must give 0. 2 lots of 2 cubed is 16, and then 11 lots of 2 squared is 44, and then we've got a times 2 which is 2a, then plus b equals 0. If we add the 16 and the 44 we get 60, so 60 plus 2a plus b equals 0, and then if we subtract 60 from both sides we could write this as 2a plus b equals negative 60. Now if we do the same thing for f of negative 6, f of negative 6 equals 0, So 2 lots of negative 6 cubed plus 11 lots of negative 6 squared plus a lots of negative 6 plus b equals 0. 2 lots of negative 6 cubed is negative 432. 11 lots of negative 6 squared is plus 396. Then we've got a lots of negative 6, so negative 6a, then plus b equals 0. If you do negative 432 add 396, you get negative 36. And then take 6a plus b equals 0. and then if you add 36 to both sides you get negative 6a plus b equals 36. This results in two simultaneous equations that we can solve. If we just subtract the equations then 2a take away 6a is 8a, b take b is 0, and negative 60 take 36 is negative 96. Divide both sides by 8 and you get a equals negative 12. If we substitute negative 12 back into the first equation We've got 2 lots of negative 12 plus b equals negative 60. 2 lots of negative 12 is negative 24 plus b is negative 60. And then add 24 to both sides and you'll get b equals negative 36. So a is negative 12 and b is negative 36. For part b we need to solve f of x equals 0, but remember we now know the values of a and b. So the function is 2x cubed plus 11x squared then minus 12x minus 36. So we need to solve this equal to zero. Now we already know two of the factors because they're given in the question. We've got x minus 2 and x plus 6. There must be one more factor and this equals zero. Now we don't need to use long division for this, we can do this quite quickly. Since when we times the first term in each bracket we must get 2x cubed. So this x times this x times whatever goes here must make 2x cubed. x times x is x squared so we're missing 2x. so this is 2x. In a similar way when we times all of the final terms we must get negative 36. So negative 2 times positive 6 times whatever goes here gives you negative 36. Negative 2 times 6 is negative 12 so we're missing a positive 3. Now that we factorize that we can see the solutions, the first bracket gives x equals 2, the next one x equals negative 6, and the final one x is negative 3 over 2. Thank you for watching this video, I hope you found it useful. Check out what I think you should watch next and also subscribe so you don't miss out on future videos.

To do that we'll substitute in negative 4. So f of negative 4 is negative 4 cubed plus 5 lots of negative 4 squared plus 2 lots of negative 4 take away 8. Since it's a show that we need to do all of the steps. So negative 4 cubed is negative 64, negative 4 squared is 16, and then times that by 5 gives you positive 80, 2 times negative 4 is negative 8, and then negative 8. And if you do this you get 0, which shows that x plus 4 is a factor. For part b we need to factorise f of x, so what we'll do is we'll take the polynomial and then we'll divide it by the factor x plus 4. So x cubed divided by x is x squared, then multiply this back, x squared times x is x cubed, and then x squared times 4 is plus 4x squared. Now we subtract these, the x cubes will cancel, 5x squared take 4x squared is just 1x squared.

Bring down the next term, and then we'll do x squared divided by x, which is just x, multiply that back, x times x is x squared and x times 4 is plus 4x. Then subtract these, the x squared will cancel, and 2x take 4x is negative 2x. Bring down the final term and then do negative 2x divided by x which is negative 2 and then multiply this back through gives you negative 2x subtract 8. Subtract these and you get 0. So we found that this could be expressed as x plus 4 times by a quadratic x squared plus x take away 2. But this quadratic can also factorize to x plus 2 x minus 1. For this question we need to show that 2x minus 1 is a factor.

To do this we'll do f of 1 half. So we need to do 2 lots of 1 half cubed, plus 13 lots of 1 half squared, plus 13 lots of 1 half, take away 10. Now 1 half cubed is 1 eighth, so we've got 2 lots of 1 eighth. 1 half squared is a quarter, so we've got 13 lots of a quarter, and then 13 lots of a half, and then take away 10. So 2 times 1 eighth is 2 eighths, which would simplify to 1 quarter.

Then we've got 13 quarters. 13 halves and take away 10. Now it would make sense to write all of these over 4, so the 13 halves will become 26 over 4 and the negative 10 will become negative 40 over 4. Now it should become clear that this equals 0, so we've finished. For part b we need to factorise, so what we'll do is take the polynomial and divide it by the factor. So 2x cubed divided by 2x is x squared, multiply this back through, We get 2x cubed, take away x squared.

Now we subtract these, the 2x cubed will cancel, and 13x squared take away 1x squared is 14x squared. Now bring down the next term, 14x squared divided by 2x gives you 7x, and then multiply this back through, 7x times 2x is 14x squared, and 7x times negative 1 is negative 7x. Now subtract these, the 14x squareds will cancel, and 13x take away negative 7x is 20x.

Then bring down the final term, 20x divided by 2x is plus 10, and multiply this back through, gives you 20x take away 10, which when you subtract these two, gives you 0. So we've found that this is the linear part times the quadratic part, and this quadratic part can further factorise to x plus 5 x plus 2. For this one we need to show that x plus 2 is a factor, so we'll do f of negative 2, which is negative 2 cubed, take away 5 lots of negative 2 squared, take away 2 lots of negative 2, add 24. Negative 2 cubed is negative 8, negative 2 squared is 4, and then times negative 5 is negative 20, negative 2 times negative 2 is positive 4, and then plus 24. And if you do this you get 0, so x plus 2 must be a factor. For part b of this question we need to solve f of x equals zero. To do this we need to fully factorize f of x. Now we have one of the factors so we'll just take f of x and divide it by that factor.

So we do x cubed divided by x which is x squared, then multiply the x squared back through, x squared times x is x cubed and x squared times 2 is plus 2x squared. Then we subtract these, x cubed take x cubed to zero, Minus 5x squared, take away 2x squared, is a negative 7x squared. Then bring down the next term and we'll do negative 7x squared divided by x, which is negative 7x, and multiply this back through.

Negative 7x times x is negative 7x squared, and negative 7x times positive 2 is negative 14x. Now subtract these, negative 7x squared will cancel, and negative 2x subtract negative 14x is 12x. Then bring down the final term, Do 12x divided by x which is plus 12 and then multiply this back through, we'll get you 12x plus 24 and you subtract these and you'll get 0. So we can now factorise this, we have x plus 2 times by the quadratic and remember it equals 0 because we're solving the equation now. We can factorise this further, the quadratic will factorise to x minus 3 x minus 4 and now we can see our solutions. So the first bracket gives us x equals negative 2 the second bracket x equals 3, and the final bracket x equals 4. For part a of this question we need to show that 4x plus 1 is a factor.

To do this we'll do f of negative a quarter. So that'll be 4 lots of negative a quarter cubed, take away 11 lots of negative a quarter squared, plus 5 lots of negative a quarter, plus 2. Now negative a quarter cubed is negative 1 over 64, so we've got 4 lots of negative 1 over 64, Negative a quarter squared is positive 1 over 16, so it's take away 11 lots of 1 16th, and then we've got plus 5 lots of negative a quarter plus 2. If we do 4 lots of negative 1 over 64, we get negative 4 over 64, which simplifies to negative a 16th, and then we do negative 11 over 16, take away 5 over 4, plus 2. Now we need all of these to be over 16, so the last two terms will change. Negative 5 over 4. would become negative 20 over 16 and the plus 2 will become 32 over 16. Now it's clear that this equals 0 therefore 4x plus 1 is a factor. For part b we need to solve f of x equals 0. To do this we need to fully factorize it. So we'll take the polynomial and divide it by the factor that we have.

4x cubed divided by 4x is x squared. Multiply this x squared back through you get 4x cubed plus x squared. Then we make a subtraction 4x cubes will cancel, negative 11x squared take away x squared is negative 12x squared.

Bring down the next term, and then do negative 12x squared divided by 4x, which gives you negative 3x, then multiply this back through, and you'll get negative 12x squared take away 3x, and then do a subtraction, the negative 12x squareds will cancel, and then 5x take away negative 3x gives you 8x. Bring down the final term, 8x divided by 4x is just plus 2, and then multiply this back through and you'll get 8x plus 2 which when you subtract gives you 0. So we've now been able to factorize this so we could write it as 4x plus 1 times y our quadratic equals 0. The quadratic will factorize further so that could be written as x take away 1 x take away 2 and then we get our solutions. The first bracket will give us x equals negative a quarter, the second bracket gives us x equals 1 and the final bracket gives us x equals 2. For this question we have to share x minus 2 as a factor so we'll do f of 2. So that's 3 lots of 2 cubed, take away 10 lots of 2 squared, plus 4 lots of 2, plus 8. 2 cubed is 8, so 3 eighths are 24, 2 squared is 4, and negative 10 times 4 is negative 40, 4 twos are 8, and then plus 8. This gives you 0, so we've shown it's a factor.

For part b we need to solve f of x equals 0. To do this we need to factorise it fully, so again we'll take our polynomial and divide it by the linear factor. 3x cubed divided by x is 3x squared. Multiply this back through and you get 3x cubed, take away 6x squared.

Now we do a subtraction, the 3x cubed will cancel, and negative 10x squared take away negative 6x squared gives you negative 4x squared. Bring down the next term, negative 4x squared divided by x gives you negative 4x, multiply this back through, negative 4x squared plus 8x. Now we subtract, the negative 4x squared will cancel, so 4x take 8x is negative 4x, bring down this final term and then do negative 4x divided by x, which gets you negative 4, and multiply that negative 4 back through, and you'll get negative 4x plus 8, take away these and you get 0. So we can write f of x as the linear part, x minus 2, times this quadratic part.

which equals 0. This quadratic part can also factorise further, so it would be 3x plus 2, x take away 2. So for this one the first bracket gives you x equals 2, the second bracket gives you x equals negative 2 thirds, and the final bracket gives you a repeated solution x equals 2 again so we don't need to write that down. In this question we're told that x plus 3 is a factor, this means that if we do f of negative 3 we know we get 0. So let's substitute that in, so negative 3 cubed plus a lots of negative 3 squared take away 21 lots of negative 3 take 18. Now we know this must equal 0, so if we simplify the left hand side, negative 3 cubed is negative 27, negative 3 squared is positive 9 and then times a gives us plus 9a, negative 21 times negative 3 is plus 63 and then take 18 equals 0. If we collect up these constant terms we end up with 9a and then plus 18 equals 0. If we rearrange this we get 9a equals negative 18. and then divide both sides by 9, we'll get a equals negative 2. And now to factorise this we'll do long division, so we'll take the polynomial, remember we now know the value of a that was negative 2, so it's negative 2 x squared, and we divide this by the linear factor. So x cubed divided by x is x squared, multiply this back through and you'll get x cubed plus 3 x squared, if you subtract these the x cubes cancel, Negative 2x squared take away 3x squared is negative 5x squared. Then we bring down the next term and negative 5x squared divided by x gives you negative 5x. And then we multiply this back through.

Negative 5x squared take away 15x. Now we subtract these, the negative 5x squareds will cancel. Negative 21x take away negative 15x gives you negative 6x.

And bring down the final term. Negative 6x divided by x gives you negative 6. and then multiply it back through, then you get negative 6x take 18, which of course when you subtract gives you 0. So we've got x plus 3 times by the quadratic factor, and then we can factorise this quadratic one into x minus 6 x plus 1. Here we have the equation of a graph along with the sketch of it. We're asked to find the value of q. Now if you substitute x equals 0 into this, All of the terms with x will disappear so we're just left with y equals q. Now when x equals 0 we're on the y-axis and this graph crosses the y-axis at negative 20, therefore q must be negative 20. So we have a graph of this equation and notice I've updated q to be negative 20 and we need to find the value of p.

Notice the graph crosses the x-axis at the number 5 here. This means if you substitute 5 in you'll get 0 out. And if this is the case by the factor theorem, x minus 5 must be a factor.

So we know that f of 5 must equal 0. So if we substitute 5 in, we get 5 cubed minus 5 squared plus p lots of 5 take 20 equals 0. We can simplify this a bit, so 5 cubed is 125, 5 squared is 25 so take 25, and then plus 5p minus 20 equals 0. If we collect up the constant terms we end up with 5p plus 80 equals 0. Take 80 from both sides and you'll get 5p equals negative 80 and then divide by 5 and you'll find that p is negative 16. We now know the values of p and q so we know the full equation of the graph. We also know one of the factors x minus 5. So if we take our graph and divide it by x minus 5 we'll find out the other factors. So x cubed divided by x is x squared, multiply this back through and you get x cubed minus 5x squared.

Do a subtraction, the x cubes cancel, negative x squared take away negative 5x squared gives you 4x squared. Bring down the next term and do 4x squared divided by x which gives you 4x. Now multiply this back through, 4x times x minus 5 gives you 4x squared take away 20x. Now subtract, the 4x squareds will cancel.

And negative 16x take away negative 20x, gives you 4x, and bring down the next term. Now 4x divided by x gives you plus 4, and multiply this 4 back through, gives you 4x take 20, and if you subtract these you get 0. So we've shown that this graph could be written in the form x minus 5, x squared plus 4x plus 4. But this can also factorise into x plus 2, x plus 2. Notice we get a repeated root here, with x plus 2 coming up. up twice.

If we were to solve the equation f of x equals zero, we would get the solutions 5, which we know, and negative 2. This means the point C here is at. In the question we were only asked for the x coordinate of the point C, so it's. In this question we're told that x-2 and x plus 6 are both factors, in which case if we do f, we should get 0. So if we do f, that's 2 lots of 2 cubed, plus 11 lots of 2 squared, plus a lots of 2, plus b, and this must give 0. 2 lots of 2 cubed is 16, and then 11 lots of 2 squared is 44, and then we've got a times 2 which is 2a, then plus b equals 0. If we add the 16 and the 44 we get 60, so 60 plus 2a plus b equals 0, and then if we subtract 60 from both sides we could write this as 2a plus b equals negative 60. Now if we do the same thing for f of negative 6, f of negative 6 equals 0, So 2 lots of negative 6 cubed plus 11 lots of negative 6 squared plus a lots of negative 6 plus b equals 0. 2 lots of negative 6 cubed is negative 432. 11 lots of negative 6 squared is plus 396. Then we've got a lots of negative 6, so negative 6a, then plus b equals 0. If you do negative 432 add 396, you get negative 36. And then take 6a plus b equals 0. and then if you add 36 to both sides you get negative 6a plus b equals 36. This results in two simultaneous equations that we can solve. If we just subtract the equations then 2a take away 6a is 8a, b take b is 0, and negative 60 take 36 is negative 96. Divide both sides by 8 and you get a equals negative 12. If we substitute negative 12 back into the first equation We've got 2 lots of negative 12 plus b equals negative 60. 2 lots of negative 12 is negative 24 plus b is negative 60. And then add 24 to both sides and you'll get b equals negative 36. So a is negative 12 and b is negative 36. For part b we need to solve f of x equals 0, but remember we now know the values of a and b.

So the function is 2x cubed plus 11x squared then minus 12x minus 36. So we need to solve this equal to zero. Now we already know two of the factors because they're given in the question. We've got x minus 2 and x plus 6. There must be one more factor and this equals zero. Now we don't need to use long division for this, we can do this quite quickly.

Since when we times the first term in each bracket we must get 2x cubed. So this x times this x times whatever goes here must make 2x cubed. x times x is x squared so we're missing 2x.

so this is 2x. In a similar way when we times all of the final terms we must get negative 36. So negative 2 times positive 6 times whatever goes here gives you negative 36. Negative 2 times 6 is negative 12 so we're missing a positive 3. Now that we factorize that we can see the solutions, the first bracket gives x equals 2, the next one x equals negative 6, and the final one x is negative 3 over 2. Thank you for watching this video, I hope you found it useful. Check out what I think you should watch next and also subscribe so you don't miss out on future videos.

Transcript for:Understanding Factor Theorem Solutions

Transcript for:
Understanding Factor Theorem Solutions