Sure! Let's find the quadratic polynomial whose zeroes are 5 and -3.
### Step-by-Step Solution:
1. Understand the Form of a Quadratic Polynomial with Given Zeroes:
A quadratic polynomial with zeroes at [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex] can be written in the form:
[tex]\[
f(x) = a(x - \alpha)(x - \beta)
\][/tex]
where [tex]\( a \)[/tex] is a non-zero constant.
2. Substitute the Given Zeroes:
Given the zeroes of the polynomial are [tex]\( \alpha = 5 \)[/tex] and [tex]\( \beta = -3 \)[/tex], we substitute these values into the form:
[tex]\[
f(x) = a(x - 5)(x + 3)
\][/tex]
3. Expand the Expression:
First, expand the factors [tex]\((x - 5)\)[/tex] and [tex]\((x + 3)\)[/tex]:
[tex]\[
(x - 5)(x + 3) = x^2 + 3x - 5x - 15 = x^2 - 2x - 15
\][/tex]
4. Include the Leading Coefficient [tex]\( a \)[/tex]:
Typically, for simplicity, we assume [tex]\( a = 1 \)[/tex] (unless specified otherwise). Hence, the polynomial becomes:
[tex]\[
f(x) = 1 \cdot (x^2 - 2x - 15) = x^2 - 2x - 15
\][/tex]
5. Identify the Coefficients of the Polynomial:
When we write the polynomial in the standard form [tex]\( ax^2 + bx + c \)[/tex], we can identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -2 \)[/tex]
- [tex]\( c = -15 \)[/tex]
### Conclusion:
The quadratic polynomial whose zeroes are 5 and -3 is:
[tex]\[
f(x) = x^2 - 2x - 15
\][/tex]
This polynomial accurately reflects the given zeroes and follows the standard form of a quadratic polynomial.
