Q. 390 - Intermediate Algebra

Question

In the following exercises, factor completely using the difference of squares pattern, if possible.

$25p^2 - 1$

Step-by-Step Solution

Answer

The Factored form of given polynomial is $(5p+1)(5p-1)$.

Step 1. Given Information

We are given a polynomial, 

$25p^2 - 1$

The formula used for factoring using the difference of squares pattern is,

$a^2-b^2=(a+b)(a-b)$

Step 2. Factorizing the polynomial

The given polynomial can be written as, 

$25p^2 - 1={\left(5p\right)}^2-1^2$

Using $a^2-b^2=(a+b)(a-b)$, we get

$25p^2 - 1=(5p+1)(5p-1)$

Step 3. Checking the factorization by multiplying

Multiplying the factors, we get 

$$\begin{gathered} (5p+1)(5p-1)=25p^2 - 1 \\ {\left(5p\right)}^2-5p+5p-1=25p^2 - 1 \\ 25p^2 - 1=25p^2 - 1 \\ LHS=RHS \end{gathered}$$

Hence the factorization is correct.