Polynomial Theorems Overview

Good day everyone, this is Teacher Cherry. In this video, we will learn about Remainder Theorem, Factor Theorem, and Rational Root Theorem. Let's have Remainder Theorem. What is Remainder Theorem? Remainder Theorem states that given a polynomial P of x with a divisor in the form x minus c, The remainder denoted by R of the polynomial is given by R equals P of C, where P of C is obtained by substituting the value indicator C to the polynomial. Remainder is the number left in dividing two expressions whose quotient is not exact. So, para mas maintindihan natin, i-apply natin yung remainder theorem. Let's have examples. Let's have example 1. Determine the remainder of x cubed plus 3x squared minus 4x plus 2 when divided by x plus 1. So, gamitin natin yung remainder theorem una. Kunin natin yung value indicator. Okay, what is our value indicator? The value indicator, c, equals negative 1. Bakit negative 1? Because... x minus c. So, x minus negative 1. Negative times negative 1 is positive 1. Or other way, equate natin sa 0 yung x plus 1. x plus 1 equals 0. Therefore, our x equals negative 1. So, yung c natin is negative 1. And then, the next step, is to substitute the value indicator here in our... polynomial okay so we have p of negative one and galing negative one that is the value indicator C okay so next substitute not an you negative one sex so we have negative 1 cube plus 3 times negative 1 squared minus 4 times negative 1 plus 2 so lahat At ng x, pinalitan natin ng negative 1. Then, solve natin. Negative 1 cubed is negative 1 times negative 1 positive 1 times negative 1 is negative 1. 3 times negative 1 squared is 3. Paano nangyari yun? Negative 1 squared is negative 1 times negative 1 is positive 1 times 3. That is equal to 3. Negative 4 times negative 1 is positive 4. Then, plus 2. Then, let's add. Negative 1 plus 3 is 2. 2 plus 4 is 6. Plus 2 is 8. So, kung ano yung nag-equal dito, yun yung magiging remainder. Therefore, what is the remainder? The remainder is 8. Okay, i-recheck natin using synthetic division. So, isulat natin yung mga coefficient nya. So, we have 1, 3. Negative 4 and 2. Ano ang value indicator? The value indicator is negative 1. So, ito yung ilalagay natin sa left. Then, the next step, bring down natin yung unang coefficient. That is 1. Then, 1 times negative 1 is negative 1. Then, add 3 plus negative 1 is positive 2. Then multiply 2 times negative 1, that is negative 2. Add negative 4 plus negative 2 is negative 6. Negative 6 times negative 1 is positive 6. Then add 2 plus 6, that is equal to 8. So parehas na 8 yung nakuha nating remainder. So the remainder is 8. Okay. Ulitin ko, by remainder theorem, kukunin muna yung value indicator and then isasubstitute mo doon sa x yung value indicator and then solve. Kung ano yung lumabas, yun yung magiging remainder. Pwede mong i-double check by using synthetic division. Okay, let's have another example. Okay, example 2. Find the value of k that will give a remainder of 2. 12 when x to the 4th plus kx cubed minus k plus 4 times x plus 6 is divided by x plus 2. So, ang kukunin naman natin dito, Ano kaya yung magiging value ng k para kapag dinivide by x plus 2, magkakaroon ng remainder na 12. Let's have our solution. The first step is to determine the value indicator. So, dito natin siya makukuha sa divisor na x plus 2. So, ang value indicator natin is negative 2. Para mas madali, yung x plus 2 i-equate. natin sa zero x plus 2 equals zero that is x equals negative 2 so yun yung magiging value indicator natin and then form tayo ng equation na p of negative 2 equals 12 so yung negative 2 yun yung ating value indicator kapag kinuha natin yung p of negative 2 magkakaroon sya ng remainder na 12. So, madadetermine natin ngayon yung value ng Okay, substitute natin yung value indicator ng negative 2 sa x ng ating polynomial. Okay, so we have x to the 4th, so we have negative 2 raised to the 4th plus kx cubed, so we have k times negative 2 cubed minus k plus 4 times x. So we have k plus 4, ang x natin is negative 2 plus 6 equals. equals 12. So, yun yung magiging remainder nya. 12. Then, solve. Negative 2 raised to the 4th is 16. Okay, paano nangyari yun? Negative 2 times negative 2 is positive 4. Positive 4 times negative 2 is negative 8. Negative 8 times negative 2 is 16. And then, K times negative 2 cubed. What is negative 2 cubed? Negative 2 times negative you 2 is 4, 4 times negative 2 is negative 8. Negative 8 times k is negative 8k. And then plus 2k, paano nangyari yung plus 2k? Negative k plus 4 times negative 2. So una, imumultiply natin yung k sa negative. Negative k, negative times k is negative k times negative 2 that is positive 2k. Next, negative Times positive 4 is negative 4. Times negative 2 is positive 8. Then, plus 6 equals 12. Okay, add natin yung mga magkakaparehas. Okay, 16 plus 8 is 24. Plus 6 is 30. So, we have 30. Negative 8k plus 2k is negative 6k. Equals 12. Transpose natin yung 30. Punta natin sa right side, magiging negative 6k equals 12 minus 30. Then, negative 6k equals 12 minus 30 is negative 18. And then, i-divide natin both sides by negative 6 para ma-eliminate yung negative 6. So, we have negative 6k over negative 6 equals negative 18 over negative 6. Negative 6k over negative 6 is positive k. Negative 18 divided by negative 6 is positive 3. So, ang nakuha nating value ng k is positive 3. Therefore, the value of k that will give a remainder of 12 when x to the 4th plus kx cubed minus k plus 4 times x plus 6 is divided by x plus 2 is 3. So, kung pinalitan natin yung k ng positive 3, pag dinivide natin itong polynomial sa x plus 2 ang makukuhang remainder ay 12. Ulit, ang k natin is 3. Okay, so proceed naman tayo sa factor theorem. Okay, let's have the factor theorem now. What is factor theorem? Factor theorem states that given a polynomial p of The expression x minus c is a factor of a polynomial if and only the value of p of c will yield to 0. That is, p of c equals 0. By factor theorem, we can say that the expression x minus c is a factor of a polynomial if what we got from p of c equals 0. Okay, para mas maintindihan, let's have example. Example 1, is x plus 2 a factor of x cubed minus 7x minus 6? Factor ba ng x cubed minus 7x minus 6 ang x plus 2? Okay, para malaman natin kung factor, gagamitan natin ng factor theorem. So, ang gagawin natin, kunin muna natin yung value indicator. So, x plus 2, ang value indicator niya is negative. 2. Paano nakuhin yung negative 2? x plus 2 equate natin sa 0 and then transpose natin x equals negative 2. Ang value indicator natin ngayon ay negative 2. And then kunin natin yung p of negative 2. p of negative 2 yung negative 2 isa substitute natin sa x sa ating polynomial. So we have x cube so negative 2 cube minus 7x negative 7 times negative 2 minus 6. Then, kunin natin ang cube ng negative 2, that is negative 8. Okay, paano na kuhay ang negative 8? Negative 2 times negative 2 is positive 4. Positive 4 times negative 2 is negative 8. Then, negative 7 times negative 2 is positive 14 minus 6. Negative 8 plus 14. is positive 6 minus 6 is 0. Ang p of negative 2 natin is equal sa 0. Okay, remember, pag nag-equal sa 0, ibig sabihin, yung x plus 2 ay factor ng x cubed minus 7x minus 6. Yun ang palaging tatandaan. Pag nag-equal sa 0, factor siya. Pero pag hindi nag-equal sa 0, not factor. Okay, let's have example 2. Is x plus 3 a factor of x cubed minus 7x minus 6? Okay, for our solution, kunin ulit natin yung value indicator. Paano nakuha yung negative 3? Yung x plus 3 equate natin sa 0, then transpose x equals negative 3. And then, yung negative 3 isasubstitute natin sa x sa ating polynomial. Okay, so we have P of negative 3 equals x cubed magiging negative 3 cubed minus 7x minus 7 times negative 3 minus 6. Evaluate natin. Negative 3 cubed ay equal sa negative 27. Negative 3 times negative 3 is positive 9. Positive 9 times negative 3 is negative 27. Negative 7 times negative 3 is positive 21 minus 6. Okay, then negative 27 plus 21 is negative 6 minus 6 that is equal to negative 12. Ang p of negative 3 ay equal sa negative 12. Ibig sabihin, ang x plus 3 ay hindi factor ng x cubed minus 7x minus 6. Kung nag-zero siya, factor. Since hindi siya nag-zero, x plus 3 is not a factor of x cubed minus 7x minus 6. Okay, ngayon naman ay ang rational root theorem. Let's now have rational root theorem. What is Rational Root Theorem? Rational Root Theorem states that given a polynomial a sub n times x raised to n plus a sub n minus 1 times x raised to n minus 1 plus a sub n minus 2 times x raised to n minus 2 plus a sub 1 times x plus a sub 0 the possible rational roots are of the form p over q where p is the set of all factors of a sub 0 and q is the set of all factors of a sub n. Para matutunan natin or may apply natin ang rational root theorem, let's have example. Example 1. Determine all the possible rational roots of x cubed minus 3x squared Minus 4x plus 12. Gagamitan natin ngayon ng rational root theorem. Okay. Kunin muna natin yung value ng p. Ang sabi dito, p is the set of all factors of a sub 0. So, ang a sub 0 natin yung ating constant. That is 12. So, p equals 12. Ano naman yung q? q is the set of all factors of a sub n. Saan matatagpuan yung a sub n? Yung a sub n natin, siya yung coefficient ng ating first term, yung highest degree. So, dito natin siya matatagpuan, coefficient ng x cube, which is 1. Therefore, q is 1. Okay, ulitin natin, ang p, yun yung constant natin, which is 12. Yung q, yung coefficient ng ating highest degree, that is 1. And then, kunin natin yung mga factors, set of all factors of a sub 0 and set of all factors of a sub n. So, kukunin natin yung factors ng 12 at saka factors ng 1. So, unahin natin yung p. Okay, so we have positive, negative 1, positive, negative 2, positive, negative 3, positive, negative 4. positive negative 6 and positive negative 12. Saan nang galing yun? Factors ng 12. 1 times 12 is 12. 2 times 6 is 12. 3 times 4 is 12. Lahat ng factors ng 12. Okay, sa Q naman tayo. Set of all factors of 1. So, positive negative 1 lang. 1 times 1 is 1. And then, kukunin natin yung P over Q. Okay, so una natin, 1 over 1, so positive negative 1. Next, 2 over 1, so we have positive negative 2. And then next, 3 over 1, positive negative 3. And then next, 4 over 1, positive negative 4. Okay, next, 6 over 1, that is positive negative 6. And last, 12 over 1, positive negative. So, ibig sabihin yung P over Q, ito lahat yung possible rational roots. Therefore, the list of all possible rational roots are positive negative 1, positive negative 2, positive negative 3, positive negative 4, positive negative 6, and positive negative 12. Okay, ulitin natin paano nakuha. kunin muna natin yung P. P ay yung constant. And then, yung Q, yun yung coefficient nung ating highest degree. And then, kunin natin yung set ng all factors ng P, set ng all factors of Q. And then, kunin natin yung quotient, P over Q. So, yung makukuha natin, yun yung all possible rational roots. Okay, let's have another example. Let's have example 2. What are the possible rational zeros of 3x raised to 5 minus x squared plus 1? Okay, so ang tinatanong dito, possible rational zeros, parehas din siya sa possible rational roots. Okay, for our solution, kunin muna natin yung value ng p. Okay, so saan galing yung 1? That is our constant. And then Q, our Q is equal to 3 sa ang galing dito sa quotient ng ating highest degree which is 3. And then kunin natin yung lahat ng factors ng P. So that is positive negative 1. 1 times 1 is 1. Next, set ng all factors ng 3. So we have positive negative 1 and positive negative 3. 1 times 3 is equal. And then yung next, kunin natin yung P over Q. So P, positive negative 1 over positive negative 1 is equal to positive negative 1. Next, P over Q, positive negative 1 over positive negative 3. That is equal to positive negative 1 over 3. Okay, so ito lahat yung ating possible rational zeros. Therefore, the list of all possible rational zeros of 3x raised to 5 minus x squared plus 1 are positive 1 are positive negative 1 and positive negative 1 over 3. Okay, so, ganun maghanap ng possible rational zeros or roots using rational root Ki Orem. Okay, that's the end of our tutorial. Before I end this video, I would like to share a simple quotation. Believe in yourself and all that you are. Know that there is something inside you that is greater than any obstacle. That is from Christian D. Larson. Thank you so much. Keep safe and have a great day.

Good day everyone, this is Teacher Cherry. In this video, we will learn about Remainder Theorem, Factor Theorem, and Rational Root Theorem. Let's have Remainder Theorem. What is Remainder Theorem?

Remainder Theorem states that given a polynomial P of x with a divisor in the form x minus c, The remainder denoted by R of the polynomial is given by R equals P of C, where P of C is obtained by substituting the value indicator C to the polynomial. Remainder is the number left in dividing two expressions whose quotient is not exact. So, para mas maintindihan natin, i-apply natin yung remainder theorem.

Let's have examples. Let's have example 1. Determine the remainder of x cubed plus 3x squared minus 4x plus 2 when divided by x plus 1. So, gamitin natin yung remainder theorem una. Kunin natin yung value indicator.

Okay, what is our value indicator? The value indicator, c, equals negative 1. Bakit negative 1? Because...

x minus c. So, x minus negative 1. Negative times negative 1 is positive 1. Or other way, equate natin sa 0 yung x plus 1. x plus 1 equals 0. Therefore, our x equals negative 1. So, yung c natin is negative 1. And then, the next step, is to substitute the value indicator here in our... polynomial okay so we have p of negative one and galing negative one that is the value indicator C okay so next substitute not an you negative one sex so we have negative 1 cube plus 3 times negative 1 squared minus 4 times negative 1 plus 2 so lahat At ng x, pinalitan natin ng negative 1. Then, solve natin. Negative 1 cubed is negative 1 times negative 1 positive 1 times negative 1 is negative 1. 3 times negative 1 squared is 3. Paano nangyari yun? Negative 1 squared is negative 1 times negative 1 is positive 1 times 3. That is equal to 3. Negative 4 times negative 1 is positive 4. Then, plus 2. Then, let's add.

Negative 1 plus 3 is 2. 2 plus 4 is 6. Plus 2 is 8. So, kung ano yung nag-equal dito, yun yung magiging remainder. Therefore, what is the remainder? The remainder is 8. Okay, i-recheck natin using synthetic division.

So, isulat natin yung mga coefficient nya. So, we have 1, 3. Negative 4 and 2. Ano ang value indicator? The value indicator is negative 1. So, ito yung ilalagay natin sa left. Then, the next step, bring down natin yung unang coefficient.

That is 1. Then, 1 times negative 1 is negative 1. Then, add 3 plus negative 1 is positive 2. Then multiply 2 times negative 1, that is negative 2. Add negative 4 plus negative 2 is negative 6. Negative 6 times negative 1 is positive 6. Then add 2 plus 6, that is equal to 8. So parehas na 8 yung nakuha nating remainder. So the remainder is 8. Okay. Ulitin ko, by remainder theorem, kukunin muna yung value indicator and then isasubstitute mo doon sa x yung value indicator and then solve.

Kung ano yung lumabas, yun yung magiging remainder. Pwede mong i-double check by using synthetic division. Okay, let's have another example.

Okay, example 2. Find the value of k that will give a remainder of 2. 12 when x to the 4th plus kx cubed minus k plus 4 times x plus 6 is divided by x plus 2. So, ang kukunin naman natin dito, Ano kaya yung magiging value ng k para kapag dinivide by x plus 2, magkakaroon ng remainder na 12. Let's have our solution. The first step is to determine the value indicator. So, dito natin siya makukuha sa divisor na x plus 2. So, ang value indicator natin is negative 2. Para mas madali, yung x plus 2 i-equate.

natin sa zero x plus 2 equals zero that is x equals negative 2 so yun yung magiging value indicator natin and then form tayo ng equation na p of negative 2 equals 12 so yung negative 2 yun yung ating value indicator kapag kinuha natin yung p of negative 2 magkakaroon sya ng remainder na 12. So, madadetermine natin ngayon yung value ng Okay, substitute natin yung value indicator ng negative 2 sa x ng ating polynomial. Okay, so we have x to the 4th, so we have negative 2 raised to the 4th plus kx cubed, so we have k times negative 2 cubed minus k plus 4 times x. So we have k plus 4, ang x natin is negative 2 plus 6 equals.

equals 12. So, yun yung magiging remainder nya. 12. Then, solve. Negative 2 raised to the 4th is 16. Okay, paano nangyari yun?

Negative 2 times negative 2 is positive 4. Positive 4 times negative 2 is negative 8. Negative 8 times negative 2 is 16. And then, K times negative 2 cubed. What is negative 2 cubed? Negative 2 times negative you 2 is 4, 4 times negative 2 is negative 8. Negative 8 times k is negative 8k.

And then plus 2k, paano nangyari yung plus 2k? Negative k plus 4 times negative 2. So una, imumultiply natin yung k sa negative. Negative k, negative times k is negative k times negative 2 that is positive 2k.

Next, negative Times positive 4 is negative 4. Times negative 2 is positive 8. Then, plus 6 equals 12. Okay, add natin yung mga magkakaparehas. Okay, 16 plus 8 is 24. Plus 6 is 30. So, we have 30. Negative 8k plus 2k is negative 6k. Equals 12. Transpose natin yung 30. Punta natin sa right side, magiging negative 6k equals 12 minus 30. Then, negative 6k equals 12 minus 30 is negative 18. And then, i-divide natin both sides by negative 6 para ma-eliminate yung negative 6. So, we have negative 6k over negative 6 equals negative 18 over negative 6. Negative 6k over negative 6 is positive k.

Negative 18 divided by negative 6 is positive 3. So, ang nakuha nating value ng k is positive 3. Therefore, the value of k that will give a remainder of 12 when x to the 4th plus kx cubed minus k plus 4 times x plus 6 is divided by x plus 2 is 3. So, kung pinalitan natin yung k ng positive 3, pag dinivide natin itong polynomial sa x plus 2 ang makukuhang remainder ay 12. Ulit, ang k natin is 3. Okay, so proceed naman tayo sa factor theorem. Okay, let's have the factor theorem now. What is factor theorem?

Factor theorem states that given a polynomial p of The expression x minus c is a factor of a polynomial if and only the value of p of c will yield to 0. That is, p of c equals 0. By factor theorem, we can say that the expression x minus c is a factor of a polynomial if what we got from p of c equals 0. Okay, para mas maintindihan, let's have example. Example 1, is x plus 2 a factor of x cubed minus 7x minus 6? Factor ba ng x cubed minus 7x minus 6 ang x plus 2? Okay, para malaman natin kung factor, gagamitan natin ng factor theorem. So, ang gagawin natin, kunin muna natin yung value indicator.

So, x plus 2, ang value indicator niya is negative. 2. Paano nakuhin yung negative 2? x plus 2 equate natin sa 0 and then transpose natin x equals negative 2. Ang value indicator natin ngayon ay negative 2. And then kunin natin yung p of negative 2. p of negative 2 yung negative 2 isa substitute natin sa x sa ating polynomial. So we have x cube so negative 2 cube minus 7x negative 7 times negative 2 minus 6. Then, kunin natin ang cube ng negative 2, that is negative 8. Okay, paano na kuhay ang negative 8?

Negative 2 times negative 2 is positive 4. Positive 4 times negative 2 is negative 8. Then, negative 7 times negative 2 is positive 14 minus 6. Negative 8 plus 14. is positive 6 minus 6 is 0. Ang p of negative 2 natin is equal sa 0. Okay, remember, pag nag-equal sa 0, ibig sabihin, yung x plus 2 ay factor ng x cubed minus 7x minus 6. Yun ang palaging tatandaan. Pag nag-equal sa 0, factor siya. Pero pag hindi nag-equal sa 0, not factor. Okay, let's have example 2. Is x plus 3 a factor of x cubed minus 7x minus 6? Okay, for our solution, kunin ulit natin yung value indicator.

Paano nakuha yung negative 3? Yung x plus 3 equate natin sa 0, then transpose x equals negative 3. And then, yung negative 3 isasubstitute natin sa x sa ating polynomial. Okay, so we have P of negative 3 equals x cubed magiging negative 3 cubed minus 7x minus 7 times negative 3 minus 6. Evaluate natin.

Negative 3 cubed ay equal sa negative 27. Negative 3 times negative 3 is positive 9. Positive 9 times negative 3 is negative 27. Negative 7 times negative 3 is positive 21 minus 6. Okay, then negative 27 plus 21 is negative 6 minus 6 that is equal to negative 12. Ang p of negative 3 ay equal sa negative 12. Ibig sabihin, ang x plus 3 ay hindi factor ng x cubed minus 7x minus 6. Kung nag-zero siya, factor. Since hindi siya nag-zero, x plus 3 is not a factor of x cubed minus 7x minus 6. Okay, ngayon naman ay ang rational root theorem. Let's now have rational root theorem. What is Rational Root Theorem?

Rational Root Theorem states that given a polynomial a sub n times x raised to n plus a sub n minus 1 times x raised to n minus 1 plus a sub n minus 2 times x raised to n minus 2 plus a sub 1 times x plus a sub 0 the possible rational roots are of the form p over q where p is the set of all factors of a sub 0 and q is the set of all factors of a sub n. Para matutunan natin or may apply natin ang rational root theorem, let's have example. Example 1. Determine all the possible rational roots of x cubed minus 3x squared Minus 4x plus 12. Gagamitan natin ngayon ng rational root theorem. Okay. Kunin muna natin yung value ng p.

Ang sabi dito, p is the set of all factors of a sub 0. So, ang a sub 0 natin yung ating constant. That is 12. So, p equals 12. Ano naman yung q? q is the set of all factors of a sub n. Saan matatagpuan yung a sub n?

Yung a sub n natin, siya yung coefficient ng ating first term, yung highest degree. So, dito natin siya matatagpuan, coefficient ng x cube, which is 1. Therefore, q is 1. Okay, ulitin natin, ang p, yun yung constant natin, which is 12. Yung q, yung coefficient ng ating highest degree, that is 1. And then, kunin natin yung mga factors, set of all factors of a sub 0 and set of all factors of a sub n. So, kukunin natin yung factors ng 12 at saka factors ng 1. So, unahin natin yung p.

Okay, so we have positive, negative 1, positive, negative 2, positive, negative 3, positive, negative 4. positive negative 6 and positive negative 12. Saan nang galing yun? Factors ng 12. 1 times 12 is 12. 2 times 6 is 12. 3 times 4 is 12. Lahat ng factors ng 12. Okay, sa Q naman tayo. Set of all factors of 1. So, positive negative 1 lang. 1 times 1 is 1. And then, kukunin natin yung P over Q. Okay, so una natin, 1 over 1, so positive negative 1. Next, 2 over 1, so we have positive negative 2. And then next, 3 over 1, positive negative 3. And then next, 4 over 1, positive negative 4. Okay, next, 6 over 1, that is positive negative 6. And last, 12 over 1, positive negative.

So, ibig sabihin yung P over Q, ito lahat yung possible rational roots. Therefore, the list of all possible rational roots are positive negative 1, positive negative 2, positive negative 3, positive negative 4, positive negative 6, and positive negative 12. Okay, ulitin natin paano nakuha. kunin muna natin yung P.

P ay yung constant. And then, yung Q, yun yung coefficient nung ating highest degree. And then, kunin natin yung set ng all factors ng P, set ng all factors of Q. And then, kunin natin yung quotient, P over Q.

So, yung makukuha natin, yun yung all possible rational roots. Okay, let's have another example. Let's have example 2. What are the possible rational zeros of 3x raised to 5 minus x squared plus 1? Okay, so ang tinatanong dito, possible rational zeros, parehas din siya sa possible rational roots. Okay, for our solution, kunin muna natin yung value ng p.

Okay, so saan galing yung 1? That is our constant. And then Q, our Q is equal to 3 sa ang galing dito sa quotient ng ating highest degree which is 3. And then kunin natin yung lahat ng factors ng P.

So that is positive negative 1. 1 times 1 is 1. Next, set ng all factors ng 3. So we have positive negative 1 and positive negative 3. 1 times 3 is equal. And then yung next, kunin natin yung P over Q. So P, positive negative 1 over positive negative 1 is equal to positive negative 1. Next, P over Q, positive negative 1 over positive negative 3. That is equal to positive negative 1 over 3. Okay, so ito lahat yung ating possible rational zeros. Therefore, the list of all possible rational zeros of 3x raised to 5 minus x squared plus 1 are positive 1 are positive negative 1 and positive negative 1 over 3. Okay, so, ganun maghanap ng possible rational zeros or roots using rational root Ki Orem.

Okay, that's the end of our tutorial. Before I end this video, I would like to share a simple quotation. Believe in yourself and all that you are.

Know that there is something inside you that is greater than any obstacle. That is from Christian D. Larson. Thank you so much. Keep safe and have a great day.

Transcript for:Polynomial Theorems Overview

Transcript for:
Polynomial Theorems Overview