Question

factor ( f(x) = 2x^3 + x^2 - 71x + 140 ) into linear factors given that ( -7 ) is a
( f(x) = 2x^3 + x^2 - 71x + 140 = square )
(factor completely.)

Explanation:

Step1: Use Factor Theorem

Since \(-7\) is a root, \((x + 7)\) is a factor. We perform polynomial division or use synthetic division to divide \(f(x)=2x^{3}+x^{2}-71x + 140\) by \((x + 7)\).
Using synthetic division:
The coefficients are \(2,1,-71,140\) and the root is \(-7\).
\[

$$\begin{array}{r|rrrr} -7 & 2 & 1 & -71 & 140 \\ & & -14 & 91 & -140 \\ \hline & 2 & -13 & 20 & 0 \\ \end{array}$$

\]
So, \(f(x)=(x + 7)(2x^{2}-13x + 20)\)

Step2: Factor the quadratic

Factor \(2x^{2}-13x + 20\). We need two numbers that multiply to \(2\times20 = 40\) and add to \(-13\). The numbers are \(-5\) and \(-8\).
Rewrite the middle term: \(2x^{2}-5x-8x + 20\)
Group: \((2x^{2}-5x)+(-8x + 20)=x(2x - 5)-4(2x - 5)=(2x - 5)(x - 4)\)

Step3: Combine factors

So, \(f(x)=(x + 7)(2x - 5)(x - 4)\)

Answer:

\((x + 7)(2x - 5)(x - 4)\)