Chapter 1.3, Problem 6E

To determine

To fill: The blank in the statement “The Special Factoring Formula for perfect square is $A^2+2AB+B^2=\underset{\_}{}$. So $x^2+10x+25$ factor as $\underset{\_}{}$”.

Answer to Problem 6E

The complete statement is “The Special Factoring Formula for perfect square is $A^2+2AB+B^2=\underset{\_}{{\left(A+B\right)}^2}$. So $x^2+10x+25$ factor as $\underset{\_}{{\left(x+5\right)}^2}$”.

Explanation of Solution

The special factoring formula for perfect square is given by the formula $A^2+2AB+B^2={\left(A+B\right)}^2$.

Rewrite the given trinomial $x^2+10x+25$ as $x^2+5\cdot 2\cdot x+5^2$.

Note that $x^2+5\cdot 2\cdot x+5^2$ is of the form $A^2+2AB+B^2$ where the middle is as follows:

$2AB=2\cdot x\cdot 5$

Thus, $A=x$ and $B=5$.

Substitute $A=x$ and $B=5$ in $A^2+2AB+B^2={\left(A+B\right)}^2$ as follows:

$x^2+10x+25={\left(x+5\right)}^2$

Therefore, the Special Factoring Formula for perfect square is $A^2+2AB+B^2=\underset{\_}{{\left(A+B\right)}^2}$. So $x^2+10x+25$ factor as $\underset{\_}{{\left(x+5\right)}^2}$.