Certainly! Let's determine the quadratic polynomial whose zeroes are 5 and -3. We can construct such a polynomial using these steps:
### Step 1: Understand the Relationship Between Zeroes and Polynomial
Given zeroes [tex]\( \alpha = 5 \)[/tex] and [tex]\( \beta = -3 \)[/tex], we know that a quadratic polynomial with roots [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex] can be represented as:
[tex]\[ k(x - \alpha)(x - \beta) \][/tex]
where [tex]\( k \)[/tex] is a constant. For simplicity and without loss of generality, we can choose [tex]\( k = 1 \)[/tex].
### Step 2: Form the Polynomial Equation
Substituting the given zeroes into the polynomial form, we get:
[tex]\[ (x - 5)(x + 3) \][/tex]
### Step 3: Expand the Polynomial Expression
Next, we need to expand the product to get the standard form [tex]\(ax^2 + bx + c\)[/tex]:
[tex]\[ (x - 5)(x + 3) = x^2 + 3x - 5x - 15 \][/tex]
### Step 4: Simplify the Polynomial Expression
Combine like terms:
[tex]\[ x^2 + 3x - 5x - 15 = x^2 - 2x - 15 \][/tex]
### Conclusion: Present the Quadratic Polynomial
The quadratic polynomial with zeroes 5 and -3 is:
[tex]\[ x^2 - 2x - 15 \][/tex]
### Coefficients
If we identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] for the polynomial [tex]\( ax^2 + bx + c \)[/tex]:
[tex]\[ a = 1, \quad b = -2, \quad c = -15 \][/tex]
Thus, the quadratic polynomial whose zeroes are 5 and -3 is given by:
[tex]\[ x^2 - 2x - 15 \][/tex]
