Question

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 _______.

Short answer

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 $x^2+7x+10=\left(x+5\right)\left(x+2\right)$

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

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