Zeros of a Quadratic Polynomial

Video Explanation

Question

Find the zeros of the quadratic polynomial \[ f(x) = x^2 – 2x – 8 \] and verify the relationship between the zeros and their coefficients.

Solution

Step 1: Find the Zeros of the Polynomial

Given:

\[ f(x) = x^2 – 2x – 8 \]

Factorising the polynomial:

\[ x^2 – 2x – 8 = x^2 – 4x + 2x – 8 \]

\[ = x(x – 4) + 2(x – 4) \]

\[ = (x – 4)(x + 2) \]

Equating each factor to zero:

\[ x – 4 = 0 \Rightarrow x = 4 \]

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

Hence, the zeros of the polynomial are:

\[ \alpha = 4,\quad \beta = -2 \]

Step 2: Verify the Relationship Between Zeros and Coefficients

For a quadratic polynomial:

\[ ax^2 + bx + c \]

Sum of zeros:

\[ \alpha + \beta = -\frac{b}{a} \]

Product of zeros:

\[ \alpha \beta = \frac{c}{a} \]

Here, \[ a = 1,\; b = -2,\; c = -8 \]

Sum of the zeros

\[ \alpha + \beta = 4 + (-2) = 2 \]

\[ -\frac{b}{a} = -\frac{-2}{1} = 2 \]

\[ \alpha + \beta = -\frac{b}{a} \quad \checkmark \]

Product of the zeros

\[ \alpha \beta = 4 \times (-2) = -8 \]

\[ \frac{c}{a} = \frac{-8}{1} = -8 \]

\[ \alpha \beta = \frac{c}{a} \quad \checkmark \]

Conclusion

The zeros of the given quadratic polynomial are:

\[ 4 \text{ and } -2 \]

The relationship between the zeros and the coefficients is verified.

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