Quadratic Polynomial from Given Zeros

Video Explanation

Question

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

\[ \text{Sum} = \frac{21}{8}, \quad \text{Product} = \frac{5}{16} \]

Also, find the zeros of this polynomial by factorisation.

Solution

Step 1: Write the General Form

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

\[ x^2 – Sx + P \]

Step 2: Form the Required Polynomial

Substitute the given values:

\[ x^2 – \frac{21}{8}x + \frac{5}{16} \]

Multiply the whole polynomial by 16 to remove fractions:

\[ 16x^2 – 42x + 5 \]

Hence, the required quadratic polynomial is:

\[ \boxed{16x^2 – 42x + 5} \]

Step 3: Factorise the Polynomial

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

\[ 16 \times 5 = 80 \]

Split the middle term using \(-40\) and \(-2\):

\[ 16x^2 – 40x – 2x + 5 \]

Grouping the terms:

\[ (16x^2 – 40x) – (2x – 5) \]

\[ 8x(2x – 5) – 1(2x – 5) \]

\[ (8x – 1)(2x – 5) \]

Step 4: Find the Zeros

\[ (8x – 1)(2x – 5) = 0 \]

\[ 8x – 1 = 0 \Rightarrow x = \frac{1}{8} \]

\[ 2x – 5 = 0 \Rightarrow x = \frac{5}{2} \]

Conclusion

The required quadratic polynomial is:

\[ \boxed{16x^2 – 42x + 5} \]

The zeros of the polynomial are:

\[ \frac{1}{8} \quad \text{and} \quad \frac{5}{2} \]

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