To factor the trinomial [tex]\( w^2 + 10w + 16 \)[/tex], follow these steps:
1. Identify the coefficients:
- The coefficient of [tex]\( w^2 \)[/tex] is 1.
- The coefficient of [tex]\( w \)[/tex] is 10.
- The constant term is 16.
2. Find two numbers that multiply to the constant term (16) and add up to the coefficient of [tex]\( w \)[/tex] (10). These two numbers are:
- Multiply to [tex]\( 16 \)[/tex]
- Add up to [tex]\( 10 \)[/tex]
After examining the factors of 16, we find:
- [tex]\( 2 \times 8 = 16 \)[/tex]
- [tex]\( 2 + 8 = 10 \)[/tex]
3. Write the trinomial as a product of two binomials:
Using the numbers identified in step 2, we can write:
[tex]\[
(w + 2)(w + 8)
\][/tex]
4. Verify the factorization:
Expand the product to ensure it matches the original trinomial:
[tex]\[
(w + 2)(w + 8) = w \cdot w + w \cdot 8 + 2 \cdot w + 2 \cdot 8 = w^2 + 8w + 2w + 16 = w^2 + 10w + 16
\][/tex]
The factorization is confirmed to be correct. Thus, the trinomial [tex]\( w^2 + 10w + 16 \)[/tex] factors to:
[tex]\[
(w + 2)(w + 8)
\][/tex]
