Quadratic Polynomial from Given Zeros

Video Explanation

Question

Find a quadratic polynomial whose sum and product of the zeros are

\[ \text{Sum} = -\frac{8}{3}, \quad \text{Product} = \frac{4}{3} \]

Also, find the zeros of this polynomial by factorisation.

Solution

Step 1: Write the General Form of the Quadratic Polynomial

If the sum of zeros is \(S\) and the product is \(P\), then the quadratic polynomial is:

\[ x^2 – Sx + P \]

Here,

\[ S = -\frac{8}{3}, \quad P = \frac{4}{3} \]

Step 2: Form the Required Polynomial

\[ x^2 – \left(-\frac{8}{3}\right)x + \frac{4}{3} \]

\[ x^2 + \frac{8}{3}x + \frac{4}{3} \]

Multiplying the entire polynomial by 3 to remove fractions:

\[ 3x^2 + 8x + 4 \]

Hence, the required quadratic polynomial is:

\[ \boxed{3x^2 + 8x + 4} \]

Step 3: Find the Zeros by Factorisation

Factorising:

\[ 3x^2 + 8x + 4 \]

Product of coefficient of \(x^2\) and constant term:

\[ 3 \times 4 = 12 \]

Split the middle term using 6 and 2:

\[ 3x^2 + 6x + 2x + 4 \]

Grouping the terms:

\[ (3x^2 + 6x) + (2x + 4) \]

\[ 3x(x + 2) + 2(x + 2) \]

\[ (3x + 2)(x + 2) \]

Step 4: Find the Zeros

\[ (3x + 2)(x + 2) = 0 \]

\[ 3x + 2 = 0 \Rightarrow x = -\frac{2}{3} \]

\[ x + 2 = 0 \Rightarrow x = -2 \]

Conclusion

The required quadratic polynomial is:

\[ \boxed{3x^2 + 8x + 4} \]

The zeros of the polynomial are:

\[ -2 \quad \text{and} \quad -\frac{2}{3} \]

\[ \therefore \quad \text{The required result is obtained.} \]