Certainly! Let's work through the problem step-by-step.
Given the zeros of the function:
- [tex]\( x = -4 \)[/tex]
- [tex]\( x = 3 \)[/tex]
### Factored Form
1. To form the quadratic expression in factored form, we start by using the given zeros to write the factors of the function.
2. A zero, [tex]\( x = a \)[/tex], of a polynomial corresponds to a factor of [tex]\( (x - a) \)[/tex].
Therefore, for the zeros provided:
- For [tex]\( x = -4 \)[/tex], the factor is [tex]\( (x - (-4)) = (x + 4) \)[/tex]
- For [tex]\( x = 3 \)[/tex], the factor is [tex]\( (x - 3) \)[/tex]
3. The quadratic expression in factored form is the product of these factors:
[tex]\[ (x + 4)(x - 3) \][/tex]
### Standard Form
1. Next, we need to convert this factored form into the standard form of the quadratic expression, which is typically written as [tex]\( ax^2 + bx + c \)[/tex].
2. To do this, we'll expand the factored form [tex]\( (x + 4)(x - 3) \)[/tex] by applying the distributive property (also known as FOIL method for binomials):
[tex]\[
(x + 4)(x - 3) = x(x - 3) + 4(x - 3)
\][/tex]
3. Distribute [tex]\( x \)[/tex] and [tex]\( 4 \)[/tex]:
[tex]\[
x(x - 3) + 4(x - 3) = x^2 - 3x + 4x - 12
\][/tex]
4. Combine like terms:
[tex]\[
x^2 - 3x + 4x - 12 = x^2 + x - 12
\][/tex]
So, the quadratic expression in standard form is:
[tex]\[ x^2 + x - 12 \][/tex]
### Summary
- Factored form: [tex]\((x + 4)(x - 3)\)[/tex]
- Standard form: [tex]\(x^2 + x - 12\)[/tex]
These forms represent the same quadratic function with the given zeros [tex]\( x = -4 \)[/tex] and [tex]\( x = 3 \)[/tex].
