Verification of Zeros of a Cubic Polynomial

Video Explanation

Question

Verify that the numbers

\[ \frac{1}{2},\; 1,\; -2 \]

are the zeroes of the cubic polynomial

\[ f(x) = 2x^3 + x^2 – 5x + 2, \]

and verify the relationship between the zeroes and coefficients.

Solution

Step 1: Verification of Zeros

A number \(x = \alpha\) is a zero of the polynomial if \(f(\alpha) = 0\).

(i) For \(x = \frac{1}{2}\)

\[ f\!\left(\frac{1}{2}\right) = 2\left(\frac{1}{2}\right)^3 + \left(\frac{1}{2}\right)^2 – 5\left(\frac{1}{2}\right) + 2 \]

\[ = 2\left(\frac{1}{8}\right) + \frac{1}{4} – \frac{5}{2} + 2 = \frac{1}{4} + \frac{1}{4} – \frac{5}{2} + 2 \]

\[ = \frac{1}{2} – \frac{5}{2} + 2 = 0 \]

Hence, \( \frac{1}{2} \) is a zero of the polynomial.

(ii) For \(x = 1\)

\[ f(1) = 2(1)^3 + (1)^2 – 5(1) + 2 = 2 + 1 – 5 + 2 = 0 \]

Hence, \(1\) is a zero of the polynomial.

(iii) For \(x = -2\)

\[ f(-2) = 2(-2)^3 + (-2)^2 – 5(-2) + 2 \]

\[ = 2(-8) + 4 + 10 + 2 = -16 + 16 = 0 \]

Hence, \(-2\) is a zero of the polynomial.

Step 2: Verification of Relationships Between Zeros and Coefficients

Let the zeroes of the polynomial be \[ \alpha = \frac{1}{2},\; \beta = 1,\; \gamma = -2. \]

The given polynomial is \[ 2x^3 + x^2 – 5x + 2. \]

Here, \[ a = 2,\quad b = 1,\quad c = -5,\quad d = 2. \]

(i) Sum of the zeroes

\[ \alpha + \beta + \gamma = \frac{1}{2} + 1 – 2 = -\frac{1}{2} \]

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

Hence, the relation \( \alpha + \beta + \gamma = -\frac{b}{a} \) is verified.

(ii) Sum of the products of zeroes taken two at a time

\[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{1}{2}(1) + 1(-2) + (-2)\left(\frac{1}{2}\right) \]

\[ = \frac{1}{2} – 2 – 1 = -\frac{5}{2} \]

\[ \frac{c}{a} = \frac{-5}{2} \]

Hence, the relation \[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \] is verified.

(iii) Product of the zeroes

\[ \alpha\beta\gamma = \frac{1}{2} \cdot 1 \cdot (-2) = -1 \]

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

Hence, the relation \[ \alpha\beta\gamma = -\frac{d}{a} \] is verified.

Conclusion

The numbers \[ \frac{1}{2},\; 1,\; -2 \] are the zeroes of the cubic polynomial \[ 2x^3 + x^2 – 5x + 2. \]

All the relationships between zeroes and coefficients are verified.