Question

It is given that (x − 2) is a factor of polynomial 2x³ − 7x² + kx − 2.

Find:

1. the value of ‘k’. 
2. Hence, factorise the resulting polynomial completely.

Answer 1

Given:

(x − 2) is a factor of f(x) = `2x^3 − 7x^2 + kx − 2`    ...[Use the Factor Theorem.]

Step 1: Use the factor theorem: if (x − 2) is a factor, then f(2) = 0.

Step 2: Evaluate `f(2) f(2)`

= `2(2)^3 − 7(2)^2 + k(2) − 2`

= 16 − 28 + 2k − 2

= − 14 + 2k

Set f(2) = 0:−14 + 2k = 0

⇒ 2k = 14

⇒ k = 7

Step 3: Substitute k and divide by (x − 2) with k = 7, f(x)

= `2x^3 − 7x^2 + 7x − 2`   ...[Perform synthetic division by x = 2 (coefficients 2, −7, 7, −2)]

Bring down 2

2 × 2

= 4 → −7 + 4

= −3

2 × (−3)

= −6 → 7 + (−6)

= 1

2 × 1

= 2 → −2 + 2

= 0 (remainder 0)

Quotient = `2x^2 − 3x + 1`

Step 4: Factor the quadratic `2x^2 − 3x + 1`

= `(2x − 1)(x − 1)`

`2x^3 − 7x^2 + 7x − 2`

= (x − 2)(2x − 1)(x − 1)