Certainly! To form a polynomial given its roots, you can use the fact that if [tex]\( r \)[/tex] is a root of the polynomial, then [tex]\( (x - r) \)[/tex] is a factor of the polynomial. Here, we have roots at [tex]\( x = -7 \)[/tex], [tex]\( x = 0 \)[/tex], and [tex]\( x = 2 \)[/tex].
### Step-by-Step Solution:
1. Identify the Roots:
- The given roots of the polynomial are [tex]\( x = -7 \)[/tex], [tex]\( x = 0 \)[/tex], and [tex]\( x = 2 \)[/tex].
2. Form Factors from Roots:
- For [tex]\( x = -7 \)[/tex], the factor is [tex]\( (x + 7) \)[/tex].
- For [tex]\( x = 0 \)[/tex], the factor is [tex]\( x \)[/tex].
- For [tex]\( x = 2 \)[/tex], the factor is [tex]\( (x - 2) \)[/tex].
3. Construct the Polynomial:
- The polynomial [tex]\( P(x) \)[/tex] can be formed by multiplying these factors together:
[tex]\[
P(x) = (x + 7) \cdot x \cdot (x - 2)
\][/tex]
4. Expand the Polynomial:
- First, multiply the first two factors:
[tex]\[
(x + 7) \cdot x = x^2 + 7x
\][/tex]
- Next, multiply this result by the remaining factor [tex]\( (x - 2) \)[/tex]:
[tex]\[
(x^2 + 7x) \cdot (x - 2)
\][/tex]
5. Distribute to Expand:
- Use the distributive property to multiply:
[tex]\[
(x^2 + 7x)(x - 2) = x^2(x - 2) + 7x(x - 2)
\][/tex]
- Break it down into simpler multiplications:
[tex]\[
x^2(x - 2) = x^3 - 2x^2
\][/tex]
[tex]\[
7x(x - 2) = 7x^2 - 14x
\][/tex]
- Combine these results:
[tex]\[
x^3 - 2x^2 + 7x^2 - 14x
\][/tex]
6. Simplify:
- Combine like terms:
[tex]\[
x^3 + 5x^2 - 14x
\][/tex]
### Final Result:
The polynomial with roots at [tex]\( x = -7 \)[/tex], [tex]\( x = 0 \)[/tex], and [tex]\( x = 2 \)[/tex] is:
[tex]\[
P(x) = x^3 + 5x^2 - 14x
\][/tex]
This is an example of a polynomial equation having the specified roots.
