Let's solve the problem of finding a quadratic polynomial whose sum and product of the zeros are given as [tex]\(\frac{8}{3}\)[/tex] and [tex]\(\frac{4}{3}\)[/tex] respectively. We'll then find the zeros of this polynomial using factorization.
### Step 1: Form the quadratic polynomial
Given:
- Sum of zeros [tex]\(\alpha + \beta = \frac{8}{3}\)[/tex]
- Product of zeros [tex]\(\alpha \cdot \beta = \frac{4}{3}\)[/tex]
A general quadratic polynomial whose zeros (roots) are [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex] can be written as:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
Using the properties of the sum and product of the roots of a quadratic polynomial:
- The sum of the roots [tex]\((\alpha + \beta) = -\frac{b}{a}\)[/tex]
- The product of the roots [tex]\((\alpha \cdot \beta) = \frac{c}{a}\)[/tex]
For simplicity, let's choose [tex]\(a = 3\)[/tex] to avoid fractions.
Now,
- From the sum of the roots: [tex]\(\frac{8}{3} = -\frac{b}{3}\)[/tex]
[tex]\[ b = -3 \cdot \frac{8}{3} = -8 \][/tex]
- From the product of the roots: [tex]\(\frac{4}{3} = \frac{c}{3}\)[/tex]
[tex]\[ c = 3 \cdot \frac{4}{3} = 4 \][/tex]
So, the quadratic polynomial is:
[tex]\[ 3x^2 - 8x + 4 = 0 \][/tex]
### Step 2: Factorize the polynomial to find the zeros
To find the zeros by factorization, let's try to rewrite the quadratic polynomial:
[tex]\[ 3x^2 - 8x + 4 \][/tex]
We need to find two numbers that multiply to [tex]\(3 \cdot 4 = 12\)[/tex] and add up to [tex]\(-8\)[/tex]. The numbers are [tex]\(-6\)[/tex] and [tex]\(-2\)[/tex].
So, we can rewrite the polynomial as:
[tex]\[ 3x^2 - 6x - 2x + 4 = 0 \][/tex]
Group the terms to factor by grouping:
[tex]\[ 3x(x - 2) - 2(x - 2) = 0 \][/tex]
Factor out the common term [tex]\((x - 2)\)[/tex]:
[tex]\[ (3x - 2)(x - 2) = 0 \][/tex]
### Step 3: Solve for the zeros
Set each factor equal to zero:
[tex]\[ 3x - 2 = 0 \quad \text{or} \quad x - 2 = 0 \][/tex]
Solving these:
[tex]\[ x = \frac{2}{3} \quad \text{or} \quad x = 2 \][/tex]
### Summary
The quadratic polynomial whose sum and product of the zeros are [tex]\(\frac{8}{3}\)[/tex] and [tex]\(\frac{4}{3}\)[/tex] respectively is:
[tex]\[ 3x^2 - 8x + 4 = 0 \][/tex]
The zeros of this polynomial are:
[tex]\[ x = \frac{2}{3} \quad \text{and} \quad x = 2 \][/tex]
