Answer :
To form a polynomial with the given zeros and degree, we need to use the fact that if a polynomial has zeros [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex], then the polynomial can be represented as:
[tex]\[ f(x) = (x - a)(x - b)(x - c) \][/tex]
In this problem, the zeros are [tex]\(-1\)[/tex], [tex]\(1\)[/tex], and [tex]\(-9\)[/tex], and the degree of the polynomial is 3. Let's construct the polynomial step by step:
1. Write the polynomial in factored form using the zeros:
[tex]\[
f(x) = (x + 1)(x - 1)(x + 9)
\][/tex]
2. First, multiply the first two factors:
[tex]\[
(x + 1)(x - 1) = x^2 - 1
\][/tex]
This results from using the difference of squares formula: [tex]\((x + 1)(x - 1) = x^2 - 1\)[/tex].
3. Next, multiply the result by the third factor:
[tex]\[
(x^2 - 1)(x + 9)
\][/tex]
Distribute [tex]\(x^2 - 1\)[/tex]:
[tex]\[
= x^2(x + 9) - 1(x + 9)
\][/tex]
Simplify each term:
[tex]\[
= (x^3 + 9x^2) - (x + 9)
\][/tex]
4. Combine like terms:
[tex]\[
= x^3 + 9x^2 - x - 9
\][/tex]
Now, we have the final polynomial:
[tex]\[ f(x) = x^3 + 9x^2 - x - 9 \][/tex]
So, the correct option that corresponds to this polynomial is:
C. [tex]\( f(x) = x^3 + 9x^2 - x - 9 \)[/tex]
[tex]\[ f(x) = (x - a)(x - b)(x - c) \][/tex]
In this problem, the zeros are [tex]\(-1\)[/tex], [tex]\(1\)[/tex], and [tex]\(-9\)[/tex], and the degree of the polynomial is 3. Let's construct the polynomial step by step:
1. Write the polynomial in factored form using the zeros:
[tex]\[
f(x) = (x + 1)(x - 1)(x + 9)
\][/tex]
2. First, multiply the first two factors:
[tex]\[
(x + 1)(x - 1) = x^2 - 1
\][/tex]
This results from using the difference of squares formula: [tex]\((x + 1)(x - 1) = x^2 - 1\)[/tex].
3. Next, multiply the result by the third factor:
[tex]\[
(x^2 - 1)(x + 9)
\][/tex]
Distribute [tex]\(x^2 - 1\)[/tex]:
[tex]\[
= x^2(x + 9) - 1(x + 9)
\][/tex]
Simplify each term:
[tex]\[
= (x^3 + 9x^2) - (x + 9)
\][/tex]
4. Combine like terms:
[tex]\[
= x^3 + 9x^2 - x - 9
\][/tex]
Now, we have the final polynomial:
[tex]\[ f(x) = x^3 + 9x^2 - x - 9 \][/tex]
So, the correct option that corresponds to this polynomial is:
C. [tex]\( f(x) = x^3 + 9x^2 - x - 9 \)[/tex]
