To factor the trinomial [tex]\(3x^2 + 26x + 35\)[/tex], follow these steps:
1. Identify the coefficients: The trinomial is in the form [tex]\(ax^2 + bx + c\)[/tex], where:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = 26\)[/tex]
- [tex]\(c = 35\)[/tex]
2. Form the product of [tex]\(a\)[/tex] and [tex]\(c\)[/tex]:
[tex]\[
ac = 3 \times 35 = 105
\][/tex]
3. Find two numbers that multiply to [tex]\(ac = 105\)[/tex] and add to [tex]\(b = 26\)[/tex]: We are looking for two numbers, [tex]\(m\)[/tex] and [tex]\(n\)[/tex], such that:
[tex]\[
m \cdot n = 105 \quad \text{and} \quad m + n = 26
\][/tex]
These numbers are [tex]\(5\)[/tex] and [tex]\(21\)[/tex], since:
[tex]\[
5 \cdot 21 = 105 \quad \text{and} \quad 5 + 21 = 26
\][/tex]
4. Rewrite the middle term using [tex]\(m\)[/tex] and [tex]\(n\)[/tex]:
We rewrite the trinomial [tex]\(3x^2 + 26x + 35\)[/tex] by splitting the middle term [tex]\(26x\)[/tex] into [tex]\(5x\)[/tex] and [tex]\(21x\)[/tex]:
[tex]\[
3x^2 + 5x + 21x + 35
\][/tex]
5. Factor by grouping:
Group the terms into two pairs and factor each pair separately:
[tex]\[
(3x^2 + 5x) + (21x + 35)
\][/tex]
Factor out the greatest common factor in each group:
[tex]\[
x(3x + 5) + 7(3x + 5)
\][/tex]
6. Factor out the common binomial factor:
Notice that [tex]\((3x + 5)\)[/tex] is a common factor in both groups. Factor out [tex]\((3x + 5)\)[/tex]:
[tex]\[
(x + 7)(3x + 5)
\][/tex]
Therefore, the trinomial [tex]\(3x^2 + 26x + 35\)[/tex] factors to:
[tex]\[
(x + 7)(3x + 5)
\][/tex]
This is the factored form of the trinomial.
