Chapter 1.3, Problem 2E

To factor the trinomial x² + 7x + 10, we look for two integers whose product is __________and whose sum is __________. These integers are __________and __________, so the trinomial factors as _______________

To determine

To fill: The blank in the statement “To factor the trinomial $x^2+7x+10$, we look for two integers whose product is $\underset{\_}{}$ and whose sum is $\underset{\_}{}$. These integers are $\underset{\_}{}$ and $\underset{\_}{}$, so the trinomial factors as $\underset{\_}{}$”.

Answer to Problem 2E

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

Explanation of Solution

The given trinomial is $x^2+7x+10$.

It is known that, the general form of trinomial is $x^2+bx+c$.

To factor a trinomial choose numbers r and s such that $r+s=b$ and $rs=c$. That is,

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

In the trinomial $x^2+7x+10$, obtain two integers whose product is 10 and sum is 7.

Note that $5\times 2=10$ and $5+2=7$. That is, the two required integers are 5 and 2.

Therefore, the factorization will be as $x^2+7x+10=\left(x+5\right)\left(x+2\right)$.

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