Sure, let's solve the problem by factoring the polynomial [tex]\(3x^2 + 20x - 32\)[/tex].
To factor this polynomial, we look for two binomials that multiply together to give us the original quadratic polynomial. Let's consider the polynomial:
[tex]\[ 3x^2 + 20x - 32 \][/tex]
Step-by-step solution:
1. Identify the coefficients:
- The coefficient of [tex]\(x^2\)[/tex] (the quadratic term) is [tex]\(a = 3\)[/tex].
- The coefficient of [tex]\(x\)[/tex] (the linear term) is [tex]\(b = 20\)[/tex].
- The constant term is [tex]\(c = -32\)[/tex].
2. Factor the polynomial:
We need to find two numbers whose product is [tex]\(a \cdot c = 3 \cdot (-32) = -96\)[/tex] and whose sum is [tex]\(b = 20\)[/tex].
3. Find the roots:
The two numbers that satisfy these conditions are [tex]\(24\)[/tex] and [tex]\(-4\)[/tex] because:
- [tex]\(24 \times (-4) = -96\)[/tex]
- [tex]\(24 + (-4) = 20\)[/tex]
4. Rewrite the middle term using these factors:
Rewrite [tex]\(20x\)[/tex] as [tex]\(24x - 4x\)[/tex]:
[tex]\[ 3x^2 + 24x - 4x - 32 \][/tex]
5. Group the terms:
Group the first two terms and the last two terms:
[tex]\[ (3x^2 + 24x) + (-4x - 32) \][/tex]
6. Factor out the greatest common factor (GCF) from each group:
[tex]\[ 3x(x + 8) - 4(x + 8) \][/tex]
7. Factor out the common binomial factor [tex]\((x + 8)\)[/tex]:
[tex]\[ (x + 8)(3x - 4) \][/tex]
Therefore, the factored form of the polynomial [tex]\(3x^2 + 20x - 32\)[/tex] is:
[tex]\[
(x + 8)(3x - 4)
\][/tex]
