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Answer :
To determine which expressions are factors of a function with the given zeros, we need to understand the concept of zeros and factors in relation to a polynomial. Here's a step-by-step breakdown:
1. Identify the Zeros:
The zeros of the function are given as [tex]\( x = -5 \)[/tex], [tex]\( x = -2 \)[/tex], and [tex]\( x = 7 \)[/tex]. A zero of a function is a value for which the function equals zero.
2. Formulate the Factors:
For each zero, there corresponds a factor of the polynomial. If [tex]\( x = a \)[/tex] is a zero, then [tex]\( x - a \)[/tex] is a factor of the polynomial. Let's find the factors using this method:
- For [tex]\( x = -5 \)[/tex], the factor is [tex]\( x - (-5) = x + 5 \)[/tex].
- For [tex]\( x = -2 \)[/tex], the factor is [tex]\( x - (-2) = x + 2 \)[/tex].
- For [tex]\( x = 7 \)[/tex], the factor is [tex]\( x - 7 \)[/tex].
3. Match the Factors to the Options:
Let's match the factors [tex]\( x + 5 \)[/tex], [tex]\( x + 2 \)[/tex], and [tex]\( x - 7 \)[/tex] with the given options:
- [tex]\( x + 7 \)[/tex]: Not a factor.
- [tex]\( x + 5 \)[/tex]: Yes, it is a factor.
- [tex]\( x - 5 \)[/tex]: Not a factor.
- [tex]\( x - 7 \)[/tex]: Yes, it is a factor.
- [tex]\( x - 2 \)[/tex]: Not a factor.
- [tex]\( x + 2 \)[/tex]: Yes, it is a factor.
So, the expressions that are factors of the function are [tex]\( x + 5 \)[/tex], [tex]\( x + 2 \)[/tex], and [tex]\( x - 7 \)[/tex]. These correspond to the zeros of the polynomial given.
1. Identify the Zeros:
The zeros of the function are given as [tex]\( x = -5 \)[/tex], [tex]\( x = -2 \)[/tex], and [tex]\( x = 7 \)[/tex]. A zero of a function is a value for which the function equals zero.
2. Formulate the Factors:
For each zero, there corresponds a factor of the polynomial. If [tex]\( x = a \)[/tex] is a zero, then [tex]\( x - a \)[/tex] is a factor of the polynomial. Let's find the factors using this method:
- For [tex]\( x = -5 \)[/tex], the factor is [tex]\( x - (-5) = x + 5 \)[/tex].
- For [tex]\( x = -2 \)[/tex], the factor is [tex]\( x - (-2) = x + 2 \)[/tex].
- For [tex]\( x = 7 \)[/tex], the factor is [tex]\( x - 7 \)[/tex].
3. Match the Factors to the Options:
Let's match the factors [tex]\( x + 5 \)[/tex], [tex]\( x + 2 \)[/tex], and [tex]\( x - 7 \)[/tex] with the given options:
- [tex]\( x + 7 \)[/tex]: Not a factor.
- [tex]\( x + 5 \)[/tex]: Yes, it is a factor.
- [tex]\( x - 5 \)[/tex]: Not a factor.
- [tex]\( x - 7 \)[/tex]: Yes, it is a factor.
- [tex]\( x - 2 \)[/tex]: Not a factor.
- [tex]\( x + 2 \)[/tex]: Yes, it is a factor.
So, the expressions that are factors of the function are [tex]\( x + 5 \)[/tex], [tex]\( x + 2 \)[/tex], and [tex]\( x - 7 \)[/tex]. These correspond to the zeros of the polynomial given.
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