Verify that the numbers given alongside the cubic polynomials below are their zeros. Also, verify the relationship between the zeros and the coefficients.fx=2x3+x2-5x+2; 12,1,-2 - Sri Chaitanya Infinity Learn Best Online Courses for NCERT Solutions, CBSE, ICSE, JEE, NEET, Olympiad and Class 6 to 12

Given:

f (x) = 2 x^{3} + x^{2} - 5 x + 2

For

x = \frac{1}{2}

\Rightarrow f (\frac{1}{2}) = 2 {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{2} - 5 \times \frac{1}{2} + 2

\Rightarrow f (\frac{1}{2}) = 2 \times \frac{1}{8} + \frac{1}{4} - \frac{5}{2} + 2

\Rightarrow f (\frac{1}{2}) = \frac{1}{4} + \frac{1}{4} - \frac{5}{2} + 2

\Rightarrow f (\frac{1}{2}) = \frac{1 + 1 - 10 + 8}{4}

\Rightarrow f (\frac{1}{2}) = \frac{0}{4}

\Rightarrow f (\frac{1}{2}) = 0

Now, for

x = 1

\Rightarrow f (1) = 2 {(1)}^{3} + {(1)}^{2} - 5 \times 1 + 2

\Rightarrow f (1) = 2 \times 1 + 1 - 5 + 2

\Rightarrow f (1) = 2 + 1 - 5 + 2

\Rightarrow f (1) = 0

And, for

x = - 2

\Rightarrow f (- 2) = 2 {(- 2)}^{3} + {(- 2)}^{2} - 5 \times (- 2) + 2

\Rightarrow f (- 2) = 2 \times (- 8) + 4 + 10 + 2

\Rightarrow f (- 2) = - 16 + 4 + 10 + 2

\Rightarrow f (- 2) = 0

Hence,

x = \frac{1}{2}, 1 - 2

are the zeros of the cubic polynomial.
Now, compare the given cubic polynomial with

a x^{3} + b x^{2} + cx + d

, we get,

\Rightarrow a = 2

\Rightarrow b = 1

\Rightarrow c = - 5

\Rightarrow d = 2

And, we know, the roots of the equation are given by

α, β, γ

, which are:

\Rightarrow α = \frac{1}{2}

\Rightarrow β = 1

\Rightarrow γ = - 2

Therefore,

\Rightarrow α + β + γ = \frac{1}{2} + 1 + (- 2)

= \frac{1}{2} + 1 - 2

= \frac{1 + 2 - 4}{2}

= - \frac{1}{2}

And,

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

Hence,

α + β + γ = - \frac{b}{a}

Now,

αβ + βγ + γα = \frac{1}{2} \times 1 + 1 \times (- 2) + (- 2) \times \frac{1}{2}

= \frac{1}{2} - 2 - 1

= \frac{1 - 4 - 2}{2}

= - \frac{5}{2}

And,

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

Hence,

αβ + βγ + γα = \frac{c}{a}

Now,

αβγ = \frac{1}{2} \times 1 \times (- 2) = - 1

And,

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

Hence,

αβγ = - \frac{d}{a}

Verify that the numbers given alongside the cubic polynomials below are their zeros. Also, verify the relationship between the zeros and the coefficients.
$f (x) = 2 x^{3} + x^{2} - 5 x + 2; \frac{1}{2}, 1, - 2$