Question

Factor f(x) into linear factors given that k is a zero of f(x).Factor f(x) = 2x^3 - 3x^2 - 5x+6; k=1

Answer 1

The given function is

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

Since k = 1 is a zero of f(x), that means

That means f(x) = 0 at x = 1

Let us find the factor from x = 1

$\begin{gathered} x=1 \\ x-1=1-1 \\ x-1=0 \end{gathered}$

The factor is (x - 1)

We will divide f(x) by (x - 1) to find the other factors

$\begin{gathered} \frac{2x^3-3x^2-5x+6}{x-1}= \\ \frac{2x^3}{x}=2x^2 \end{gathered}$

Multiply 2x^2 by (x -1 ) then subtract the answer from f(x)

$\begin{gathered} 2x^2(x-1)=2x^3-2x^2 \\ 2x^3-3x^2-5x+6-(2x^3-2x^2)=-x^2-5x-6 \end{gathered}$

Divide -x^2 by x, then multiply the answer by (x - 1), then subtract the result from the answer above

$\begin{gathered} -\frac{x^2}{x}=-x \\ -x(x-1)=-x^2+x \\ -x^2-5x+6-(-x^2+x)=-6x+6 \end{gathered}$

Divide -6x by x, then multiply the result by (x - 1), then subtract the answer from the last answer above

$\begin{gathered} \frac{-6x}{x}=-6 \\ -6(x-1)=-6x+6 \\ -6x+6-(-6x+6)=0 \end{gathered}$

Then

$\frac{2x^3-3x^2-5x+6}{x-1}=2x^2-x-6$

Now we will factorize it into two factors

$\begin{gathered} 2x^2=(2x)(x) \\ -6=(-2)(3) \\ (-2)(2x)+(3)(x)=-4x+3x=-x \end{gathered}$

Then the 2 factors are (2x + 3) and (x - 2)

The factors of f(x) are

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