Question

What are the linear factors of the function f(x)=2x3−x2−13x−6? Select each correct answer.

(x+3)

(2x−1)

(x−2)

(x+2)

(2x+1)

(x−3)

Last 3 are right

Answer 1

Given:

The function is

$f(x)=2x^3-x^2-13x-6$

To find:

The linear factors of the given function.

Solution:

We have,

$f(x)=2x^3-x^2-13x-6$

Splitting the middle terms, it can be rewritten as

$f(x)=2x^3-(6-5)x^2-(15-2)x-6$

$f(x)=2x^3-6x^2+5x^2-15x+2x-6$

$f(x)=2x^2(x-3)+5x(x-3)+2(x-3)$

$f(x)=(x-3)(2x^2+5x+2)$

Now, Splitting the middle term, we get

$f(x)=(x-3)(2x^2+4x+x+2)$

$f(x)=(x-3)(2x(x+2)+1(x+2))$

$f(x)=(x-3)(x+2)(2x+1)$

Therefore, the three linear factors of given function are (x-3), (x+2) and (2x+1). Hence, option 4, 5 and 6 are correct.

Answer 2

The linear factors of the given function f(x) = 2x³ - x² - 13x - 6 are;

(x - 3), (x + 2) and (2x + 1)

We are given the function;

f(x) = 2x³ - x² - 13x - 6

Let us start by splitting the terms at the middle. Since last term is 6, we can write 1 before x² as (6 - 5)

Also, we can write 13 as (15 - 2). Thus;

f(x) = 2x³ - (6 - 5)x² - (15 - 2)x - 6

Expanding gives us;

f(x) = 2x³ - 6x² + 5x² - 15x + 2x - 6

Factorizing gives us;

f(x) = 2x²(x - 3) + 5x(x - 3) + 2(x - 3)

(x - 3) is common and so we factorize it out to get;

f(x) = (x - 3)(2x² + 5x + 2)

Let us split (2x² + 5x + 2); 5 can be expressed as 4 + 1.Thus;

f(x) = (x - 3)(2x² + (4 + 1)x + 2)

f(x) = (x - 3)(2x² + 4x + x + 2)

Factorizing again gives;

f(x) = (x - 3)(2x(x + 2) + 1(x +2))

f(x) = (x - 3)(x + 2)(2x + 1)

In conclusion, the linear factors of the given function f(x) = 2x³ - x² - 13x - 6 are; (x - 3), (x + 2) and (2x + 1)

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