Question

Factor each polynomial.

$\mathbf{5}{a}^2-\mathbf{25}a-a+\mathbf{5}$

Short answer

The factors of given polynomial are $\left(\mathbf{5}a-\mathbf{1}\right)\left(a-\mathbf{5}\right)$.

Step-by-step solution

Step 1. Combine.

Combine like terms in the given polynomial.

$5a^2-26a+5$

Step 2. Find the GCF.

Find the GCF of$5a^2,26a$and 5.

$$\begin{gathered} 5a^2=1\cdot 5\cdot a\cdot a \\ 26a=1\cdot 2\cdot 13\cdot a \\ 5=1\cdot 5 \end{gathered}$$

Therefore, GCF is 1.

Step 3. Apply factoring pattern of ax2+bx+c.

There are two numbers having product of 25 and sum of $-26$. The numbers are $-1$and $-25$.

$5a^2+mx+nx+5=5a^2-25a-1a+5$

Step 4. Group terms with common factors and factor out GCF.

Group terms with common factors and factor out GCF from each grouping.

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

Therefore, the factors of$5a^2-25a-a+5$ are $\left(5a-1\right)\left(a-5\right)$.