Factor and Remainder Theorems

Edexcel IAL WMA12/01/P2/Jan 2020/Q03a, b (The Factor Theorem, Factorizing)

f(x) = 6x3 + 17x2 + 4x – 12 

• Use the factor theorem to show that (2x + 3) is a factor of f(x).​​ 

(2)

• Hence, using algebra, write f(x) as a product of three linear factors.​​ 

(4)

SOLUTION

a-​​ 

f(x) = 6x3 + 17x2 + 4x – 12

Using the factor theorem.​​ 

If x=-32 is a root then f-32 must be equals to​​ 0.

f-32 = 6-323 + 17-322 + 4-32–12

f-32=6-278+17 94+4-32-12

f-32=0

As​​  f-32 =0,​​ 2x+3 is a factor fx.​​ 

b- Using algebraic method, meaning thereby using long division method.​​ 

3x2+4x-42x+3⟌6x3 + 17x2 + 4x – 12-6x3+9x2                           8x2+4x                            -8x2+12x                                                 -8x-12                                                 -8x-12-            -              -

6x3+17x2+4x-12=2x+3(3x2+4x-4)

Solving the quadratic equation by factorization/middle term breaking method.​​ 

3x2+4x-4=3x2+6x-2x-4

=3xx+2-2(x+2)

=(x+2)(3x-2)

Hence,​​ f(x)​​ can be written as the product of three linear factors as shown here.

6x3+17x2+4x-12=2x+3x+23x-2