To find the zeros (roots) of the polynomial [tex]\(x^2 - 2x - 8\)[/tex] and verify the relationships between the sum and product of the roots and the coefficients, follow these steps:
### Step 1: Identify the Polynomial
The given polynomial is:
[tex]\[x^2 - 2x - 8\][/tex]
### Step 2: Find the Zeros of the Polynomial
The zeros of the polynomial are the values of [tex]\(x\)[/tex] for which the polynomial equals zero:
[tex]\[x^2 - 2x - 8 = 0\][/tex]
Factor the quadratic expression:
[tex]\[x^2 - 2x - 8 = (x - 4)(x + 2) = 0\][/tex]
Set each factor equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[x - 4 = 0 \quad \Rightarrow \quad x = 4\][/tex]
[tex]\[x + 2 = 0 \quad \Rightarrow \quad x = -2\][/tex]
So, the zeros of the polynomial are:
[tex]\[x = 4 \quad \text{and} \quad x = -2\][/tex]
### Step 3: Verify the Sum of the Zeros
For a quadratic polynomial [tex]\(ax^2 + bx + c\)[/tex], the sum of the zeros is given by [tex]\(-\frac{b}{a}\)[/tex].
Here, the coefficients are:
[tex]\[a = 1, \quad b = -2, \quad c = -8\][/tex]
Sum of the zeros based on the zeros found:
[tex]\[4 + (-2) = 2\][/tex]
Sum of the zeros using the coefficient relationship:
[tex]\[-\frac{b}{a} = -\frac{-2}{1} = 2\][/tex]
Both methods give the same result, confirming that the sum of the zeros is [tex]\(2\)[/tex].
### Step 4: Verify the Product of the Zeros
For a quadratic polynomial [tex]\(ax^2 + bx + c\)[/tex], the product of the zeros is given by [tex]\(\frac{c}{a}\)[/tex].
Product of the zeros based on the zeros found:
[tex]\[4 \cdot (-2) = -8\][/tex]
Product of the zeros using the coefficient relationship:
[tex]\[\frac{c}{a} = \frac{-8}{1} = -8\][/tex]
Both methods give the same result, confirming that the product of the zeros is [tex]\(-8\)[/tex].
### Summary
The zeros of the polynomial [tex]\(x^2 - 2x - 8\)[/tex] are [tex]\(4\)[/tex] and [tex]\(-2\)[/tex]. The sum of the zeros is [tex]\(2\)[/tex], and the product of the zeros is [tex]\(-8\)[/tex]. These results are consistent with the relationships:
[tex]\[ \text{Sum of the zeros} = -\frac{b}{a} = 2 \][/tex]
[tex]\[ \text{Product of the zeros} = \frac{c}{a} = -8 \][/tex]
Thus, the zeros and their relationships to the coefficients are validated.
