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Part B: Exceeds Mastery
Given these zeros of a function, [tex]\(x = -4\)[/tex] and [tex]\(x = 3\)[/tex], find a quadratic expression as a product (Factored form) and as a sum (Standard form) to represent the function.
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]
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