We have an answer to "A polynomial and one of its factors is..."

High School
Algebra
Pre-Algebra
Factors and multiples

nicekikah

Answered question

2021-08-05

A polynomial and one of its factors is given. Factor the polynomial COMPLETELY given that one of its factors is

x + 4

. Your final answer should be in factored form.

f (x) = x^{7} + 4 x^{6} + 7 x^{4} + 28 x^{3} - 8 x - 32

Answer & Explanation

Lacey-May Snyder

Skilled2021-08-06Added 88 answers

Step 1
We use the synthetic division to get the other factor for the missing terms, we will use 0 as the coefficient.
$\begin{array}{cc} - 4 & 1 & 4 & 0 & 7 & 28 & 0 & - 8 & - 32 \\ - 4 & 0 & 0 & - 28 & 0 & 0 & 32 \\ 1 & 0 & 0 & 7 & 0 & 0 & - 8 & 0 \end{array}$
So the quotient is $x^{6} + 7 x^{3} - 8$
Step 2
Then we try to factor $x^{6} + 7 x^{3} - 8$
$x^{6} + 7 x^{3} - 8$
$= (x^{3} + 8) (x^{3} - 1)$
$= (x^{3} + 2^{3}) (x^{3} - 1^{3})$
$= (x + 2) (x^{2} - 2 x + 4) (x - 1) (x^{2} + x + 1)$
Answer: $(x + 4) (x + 2) (x^{2} - 2 x + 4) (x - 1) (x^{2} + x + 1)$

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