Let's factor the trinomial [tex]\(x^2 + 10x + 16\)[/tex] step-by-step.
Step 1: Identify the coefficients
For the trinomial [tex]\(x^2 + 10x + 16\)[/tex]:
- The coefficient of [tex]\(x^2\)[/tex] (denoted as [tex]\(a\)[/tex]) is 1.
- The coefficient of [tex]\(x\)[/tex] (denoted as [tex]\(b\)[/tex]) is 10.
- The constant term (denoted as [tex]\(c\)[/tex]) is 16.
Step 2: Calculate the product of [tex]\(a\)[/tex] and [tex]\(c\)[/tex]
Calculate [tex]\(ac\)[/tex]:
[tex]\[ ac = 1 \times 16 = 16 \][/tex]
Step 3: Find two numbers that multiply to [tex]\(ac\)[/tex] and add to [tex]\(b\)[/tex]
We need to find two numbers that:
- Multiply to 16 (the product of [tex]\(ac\)[/tex]),
- Add up to 10 (the value of [tex]\(b\)[/tex]).
The pairs of factors of 16 are:
[tex]\[
(1, 16), (2, 8), (4, 4)
\][/tex]
Among these pairs, the pair that adds up to 10 is (2, 8):
[tex]\[
2 + 8 = 10
\][/tex]
Thus, the two numbers we are looking for are 2 and 8.
Step 4: Rewrite the middle term using the two numbers found
Rewrite [tex]\(10x\)[/tex] using 2 and 8:
[tex]\[
x^2 + 10x + 16 = x^2 + 2x + 8x + 16
\][/tex]
Step 5: Factor by grouping
Group the terms to factor by grouping:
[tex]\[
x^2 + 2x + 8x + 16 = (x^2 + 2x) + (8x + 16)
\][/tex]
Factor out the greatest common factor (GCF) from each group:
[tex]\[
x(x + 2) + 8(x + 2)
\][/tex]
Step 6: Factor out the common binomial factor
Notice that [tex]\((x + 2)\)[/tex] is a common factor:
[tex]\[
x(x + 2) + 8(x + 2) = (x + 2)(x + 8)
\][/tex]
Step 7: Write the factored form
The factored form of the trinomial [tex]\(x^2 + 10x + 16\)[/tex] is:
[tex]\[
(x + 2)(x + 8)
\][/tex]
Thus, we have successfully factored the trinomial [tex]\(x^2 + 10x + 16\)[/tex] into [tex]\((x + 2)(x + 8)\)[/tex].
