Question

Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases:

(i) p(x)=2x³ + x² - 2x-1, g(x)=x+1
(ii) p(x)= x³ + 3x² + 3x + 1, g(x)=x+2
(iii) p(x)=x² - 4x²+x+6, g(x)=x-3

Can anyone plz solve these in step by step answers?

Answer 1

Step-by-step explanation:

(1) p(x) = 2x³+x²-2x-1 , g(x) = x+1

Let , g(x) = 0

=> x+1 = 0 => x = -1

Using factor theorem to check whether the p(x) is divisible by g(x)

p(-1) = 2(-1)³ + (-1)² - 2(-1) -1

p(-1) = 2(-1) + 1 + 2 -1

p(-1) = -2 +2

p(-1) = 0

Yes , g(x) is a factor of p(x)

(2) p(x)= x³ + 3x² + 3x + 1, g(x)=x+2

Let, g(x) = 0

=> x + 2 = 0 => x = -2

Using factor theorem to check whether the p(x) is divisible by g(x)

p(-2) = (-2)³ + 3(-2)² + 3(-2) + 1

p(-2) = -8 + 3(4) - 6 +1

p(-2) = -14 + 12 + 1

p(-2) = -1

No, g(x) is not a factor of p(x)

(3) x² - 4x²+x+6, g(x)=x-3

Let, g(x) = 0

=> x - 3 = 0 => x = 3

Using factor theorem to check whether the p(x) is divisible by g(x)

p(3) = (3)² - 4(3)² + 3 + 6

p(3) = 9 - 4(9) + 9

p(3) = 18 - 36

p(3) = -18

No, g(x) is not a factor of p(x)