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### Precalculus Mathematics for Calculus

Book edition 7th Edition
Author(s) James Stewart, Lothar Redlin, Saleem Watson
Pages 948 pages
ISBN 9781337067508

# To factor the trinomial ${x}^{2}+7x+10$, we look for two integers whose product is _____ and whose sum is _____. These integers are _____ and _____, so the trinomial factors as _______.

To factor the trinomial ${x}^{2}+7x+10$, we look for two integers whose product is 10 and whose sum is 7. These integers are 5 and 2 , so the trinomial factors as ${x}^{2}+7x+10=\left(x+5\right)\left(x+2\right)$

See the step by step solution

## Step 1. Apply the concept of factoring trinomials.

To factor a trinomial of the form $a{x}^{2}+bx+c$, we note that

$\left(x+r\right)\left(x+s\right)={x}^{2}+\left(r+s\right)x+rs$

So, we need to choose numbers r and s so that $r+s=b$ and $rs=c$.

## Step 2. Analyze the trinomial.

Given ${x}^{2}+7x+10$

This is of the form $a{x}^{2}+bx+c$ where $b=7$ and $c=10$

The factorization of the trinomial will be $\left(x+r\right)\left(x+s\right)={x}^{2}+\left(r+s\right)x+rs$

So, we need to choose numbers r and s so that $r+s=b$ and $rs=c$.

## Step 3. Find the integers to factor the trinomial.

We need to choose integers r and s so that $r+s=7$ and $rs=10$.

By trial and error we find that the two integers are $r=5,s=2$

$r+s=5+2=7$

$rs=5\cdot 2=10$

## Step 4. Factor the trinomial.

The factorization of the trinomial is

$\begin{array}{l}{x}^{2}+7x+10=\left(x+r\right)\left(x+s\right)\\ ⇒{x}^{2}+7x+10=\left(x+5\right)\left(x+2\right)\end{array}$