To factor the quadratic polynomial [tex]\(3x^2 + 11x - 4\)[/tex], we need to express it as a product of two binomials. Let's break down the steps:
1. Identify a quadratic polynomial:
The given polynomial is [tex]\(3x^2 + 11x - 4\)[/tex].
2. Finding the factors:
We are looking for two binomials of the form [tex]\((ax + b)(cx + d)\)[/tex] such that when multiplied together, they yield the original polynomial.
3. Given Possible Factors:
- [tex]\((3x + 1)(x - 4)\)[/tex]
- [tex]\((3x - 1)(x + 4)\)[/tex]
- [tex]\((3x - 2)(x + 2)\)[/tex]
- [tex]\((3x + 2)(x - 2)\)[/tex]
4. Check each possible factorization:
- [tex]\((3x + 1)(x - 4)\)[/tex]:
Calculate it: [tex]\((3x + 1)(x - 4) = 3x^2 - 12x + x - 4 = 3x^2 - 11x - 4\)[/tex] which does not match the polynomial.
- [tex]\((3x - 1)(x + 4)\)[/tex]:
Calculate it: [tex]\((3x - 1)(x + 4) = 3x^2 + 12x - x - 4 = 3x^2 + 11x - 4\)[/tex] which matches the polynomial.
- [tex]\((3x - 2)(x + 2)\)[/tex]:
Calculate it: [tex]\((3x - 2)(x + 2) = 3x^2 + 6x - 2x - 4 = 3x^2 + 4x - 4\)[/tex] which does not match the polynomial.
- [tex]\((3x + 2)(x - 2)\)[/tex]:
Calculate it: [tex]\((3x + 2)(x - 2) = 3x^2 - 6x + 2x - 4 = 3x^2 - 4x - 4\)[/tex] which does not match the polynomial.
Based on the above verifications, the correct factorization of the quadratic polynomial [tex]\(3x^2 + 11x - 4\)[/tex] is:
[tex]\[
(3x - 1)(x + 4)
\][/tex]
Therefore, the factors that Jenna needs to model for the sides are [tex]\((3x - 1)\)[/tex] and [tex]\((x + 4)\)[/tex].
