Question

Use the Factor Theorem and synthetic division to show x + 5 is a factor of f(x) = 2x3 + 7x2 − 14x + 5

Answer 1

Factor theorem:
`f(-5) = 0`.

Synthetic division:
`f(x) = (x + 5)\, (2\, x^2 -3\, x + 1)`.

Explanation:

Factor Theorem

By the factor theorem, a monomial of the form
`(x - a)` (where
`a` is a constant) is a factor of polynomial
`f(x)` if and only if
`f(a) = 0`.

In this question, the monomial is
`(x + 5)`, which is equivalently
`(x - (-5))`.
`a = -5`.

`\begin{aligned}& f(-5) \\ &= 2 * (-5)^(3) + 7 * (-5)^(2)- 14* (-5) + 5\\ &= -250 + 175 + 70 + 5 \\ &= 0\end{aligned}`.

Hence, by the factor theorem,
`(x + 5)`, which is equivalent to
`(x - (-5))`, is a factor of
`f(x)`.

Synthetic Division

`\begin{aligned}& f(x) \\ &= 2\, x^(3) + 7\, x^(2) - 14\, x + 5 \\ &= \underbrace{(x + 5) \, (2\, x^2)}_(2\, x^(3) + 10\, x^(2)) - 3\, x^(2) - 14\, x + 5 \\ &= \underbrace{(x + 5) \, (2\, x^2)}_(2\, x^(3) + 10\, x^(2)) + \underbrace{(x + 5)\, (-3\, x)}_(-3\, x^(2) - 15\, x) + (x + 5) \\ &= (x + 5)\, (2\, x^(2) - 3\, x + 1)\end{aligned}`.

The remainder is
`0` when dividing
`f(x)` by
`(x + 5)`. Hence,
`(x + 5)\!` is a factor of
`f(x)\!`.