Answer :
To find a polynomial equation with given zeros of 5, -3, and 3, we start by understanding what these zeros mean in terms of factors of the polynomial. If a polynomial has zeros at [tex]\(x = 5\)[/tex], [tex]\(x = -3\)[/tex], and [tex]\(x = 3\)[/tex], then it can be represented by factors as follows:
- For zero at [tex]\(x = 5\)[/tex], the factor is [tex]\((x - 5)\)[/tex].
- For zero at [tex]\(x = -3\)[/tex], the factor is [tex]\((x + 3)\)[/tex].
- For zero at [tex]\(x = 3\)[/tex], the factor is [tex]\((x - 3)\)[/tex].
The polynomial is obtained by multiplying these factors together:
[tex]\[
(x - 5)(x + 3)(x - 3)
\][/tex]
First, we handle the multiplication of [tex]\((x + 3)(x - 3)\)[/tex], which is a difference of squares:
[tex]\[
(x + 3)(x - 3) = x^2 - 9
\][/tex]
Now, multiply this result by the remaining factor [tex]\((x - 5)\)[/tex]:
[tex]\[
(x - 5)(x^2 - 9) = x(x^2 - 9) - 5(x^2 - 9)
\][/tex]
[tex]\[
= x^3 - 9x - 5x^2 + 45
\][/tex]
Rearranging the terms, we get:
[tex]\[
x^3 - 5x^2 - 9x + 45
\][/tex]
This expression suggests that the polynomial equation with zeros at 5, -3, and 3 is:
[tex]\[
x^3 - 5x^2 - 9x + 45 = 0
\][/tex]
Among the given options, this matches with:
b. [tex]\(x^3 - 5x^2 - 9x + 45 = 0\)[/tex]
Therefore, the correct option is b.
- For zero at [tex]\(x = 5\)[/tex], the factor is [tex]\((x - 5)\)[/tex].
- For zero at [tex]\(x = -3\)[/tex], the factor is [tex]\((x + 3)\)[/tex].
- For zero at [tex]\(x = 3\)[/tex], the factor is [tex]\((x - 3)\)[/tex].
The polynomial is obtained by multiplying these factors together:
[tex]\[
(x - 5)(x + 3)(x - 3)
\][/tex]
First, we handle the multiplication of [tex]\((x + 3)(x - 3)\)[/tex], which is a difference of squares:
[tex]\[
(x + 3)(x - 3) = x^2 - 9
\][/tex]
Now, multiply this result by the remaining factor [tex]\((x - 5)\)[/tex]:
[tex]\[
(x - 5)(x^2 - 9) = x(x^2 - 9) - 5(x^2 - 9)
\][/tex]
[tex]\[
= x^3 - 9x - 5x^2 + 45
\][/tex]
Rearranging the terms, we get:
[tex]\[
x^3 - 5x^2 - 9x + 45
\][/tex]
This expression suggests that the polynomial equation with zeros at 5, -3, and 3 is:
[tex]\[
x^3 - 5x^2 - 9x + 45 = 0
\][/tex]
Among the given options, this matches with:
b. [tex]\(x^3 - 5x^2 - 9x + 45 = 0\)[/tex]
Therefore, the correct option is b.
