Q. 61E - Intermediate Algebra

Question

In the following exercises, factor each trinomial of the form, $x^2+bx+c$.

$p^2+11p+30$

Step-by-Step Solution

Answer

The factors of $p^2+11p+30$ are $(p+5)(p+6)$.

Step 1. Given and explanation

We have $p^2+11p+30$.

We will first find any common factor. if present, in the equation. Then we will any two numbers whose product comes out to be the product of the first and last term of the equation and the sum comes out to be the middle term.

Then by splitting the middle term, we will factor out the common terms and will get the factors of the trinomial.

Step 2. Finding factors.

We can see that there is no common factor in the equation. SO we will proceed finding the numbers whose product comes out to be $30$ and sum results in $11$. The factors of $30$ are-

Numbers	Sums
$1 and 30$	$31$
$2 and 15$	$17$
$3 and 10$	$13$
$5 and 6$	$11$

Through this we can say that the two numbers are $5,6$.

Step 3. Splitting the middle term.

We have two numbers as $5,6$.

Splitting the middle terms gives us,

$$\begin{gathered} =p^2+11p+30 \\ =p^2+5p+6p+30 \\ =p(p+5)+6(p+5) \\ =(p+5)(p+6) \end{gathered}$$

This shows that factors are $(p+5)(p+6)$.

Step 4. Checking the solution.

We will check the solution by multiplying. If we get the same given equation, our calculations are right.

So,

$$\begin{gathered} =(p+5)(p+6) \\ =p^2+6p+5p+30 \\ =p^2+11p+30 \end{gathered}$$

Thus our calculations are right.