Solutions to Factor Theorem Questions

Introduction

• The video covers solutions to questions related to the factor theorem.
• Link to questions for practice is available in the description.

Example 1: Show x+4 is a Factor

• Objective: Prove x+4 is a factor.

• Method: Substitute x = -4 in the polynomial.

• Steps:

  • Calculate ( f(-4) = (-4)^3 + 5(-4)^2 + 2(-4) - 8 )
  • ( (-4)^3 = -64, (-4)^2 = 16 \times 5 = 80, 2 \times -4 = -8 )
  • Sum results to get 0, proving x+4 is a factor.

• Factorisation:

  • Divide polynomial by x+4.
  • Result is ( (x+4)(x^2 + x - 2) ).
  • Further factorize ( x^2 + x - 2 ) to ( (x+2)(x-1) ).

Example 2: Show 2x-1 is a Factor

• Objective: Prove 2x-1 is a factor.

• Method: Substitute x = 1/2 in the polynomial.

• Steps:

  • Calculate ( f(1/2) = 2(1/2)^3 + 13(1/2)^2 + 13(1/2) - 10 )
  • Simplify using common denominators; result equals 0.

• Factorisation:

  • Divide polynomial by 2x-1.
  • Result is linear part times quadratic part, ( x^2 + 7x + 10 ).
  • Further factorize to ( (x+5)(x+2) ).

Example 3: Show x+2 is a Factor

• Objective: Prove x+2 is a factor.

• Method: Substitute x = -2 in the polynomial.

• Steps:

  • Calculate ( f(-2) = (-2)^3 - 5(-2)^2 - 2(-2) + 24 )
  • Result equals 0.

• Solving f(x) = 0:

  • Fully factorize ( f(x) ).
  • Use division to get ( (x+2)(x^2-3x-4) ).
  • Further factorize quadratic to find solutions: ( x = -2, 3, 4 ).

Example 4: Show 4x+1 is a Factor

• Objective: Prove 4x+1 is a factor.

• Method: Substitute x = -1/4 in the polynomial.

• Steps:

  • Simplify calculations with common denominators.
  • Result equals 0.

• Solving f(x) = 0:

  • Factorize using division: ( (4x+1)(x^2-3x+2) ).
  • Further factorize: ( (x-1)(x-2) ), solutions are ( x = -1/4, 1, 2 ).

Example 5: Show x-2 is a Factor

• Objective: Prove x-2 is a factor.

• Method: Substitute x = 2 in the polynomial.

• Steps:

  • Evaluate ( f(2) ) resulting in 0.

• Solving f(x) = 0:

  • Factorize polynomial: ( (x-2)(3x^2-4x-4) ).
  • Further factorize: ( (3x+2)(x-2) ).
  • Find solutions: ( x = 2, -2/3 ) (repeated root).

Example 6: Given x+3 is a Factor

• Objective: Use given factor x+3.

• Method: Substitute x = -3.

• Steps:

  • Simplify expression to find constants, ( a = -2 ).

• Factorisation:

  • Use long division finding ( x+3 ) times quadratic.
  • Factorize quadratic to ( (x-6)(x+1) ).

Graph Equation Questions

• Finding q:

  • Substitute x=0, get y=q.
  • y-intercept is -20, so q=-20.

• Finding p:

  • Use x-axis crossing point at 5.
  • Substitute x=5 for factor theorem; solve to find p.

• Factorisation and Solutions:

  • Known factor x-5; divide to find others.
  • Resulting quadratic shows repeated roots.

Solving Simultaneous Equations with Factors

• Objective: Find a and b given factors.

• Method:

  • Set up equations from known factors.
  • Solve simultaneous equations to find a and b.

• Solving f(x) = 0:

  • With updated polynomial factors.
  • Use provided factors and additional factorisation.

Conclusion

• Recap on factor theorem utility.
• Encourage further practice and study.
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