Question

find the zeros of the cubic polynomial 4 x cube + 10 x square plus 6x and verify the relationship between the zeros and the coefficient

Answer 1

three zeros of polynomial is 0 , -1 and -3/2

Answer 2

$Let \: Cubic \: polynomial \\p(x) = 4x^{3} + 10x^{2} + 6x$

/* Compare p(x) with ax³+bx²+cx+d , we get */

$a = 4 , b = 10 , c = 6 \:and \: d = 0$

$p(x) = 4x^{3} + 10x^{2} + 6x\\= 2x(2x^{2}+5x+3)\\= 2x(2x^{2} + 2x + 3x + 3)\\= 2x[(2x(x+1)+3(x+1)]\\= 2x(x+1)(2x+3)$

$So, the \: value \: of \: 4x^{3} + 10x^{2} + 6x\\is \: zero \:when \: x = 0 \:Or \: x = -1 \\Or \: x = \frac{-3}{2}$

Therefore.,

$The \: Zeroes \:of \: 4x^{3} + 10x^{2} + 6x \:are \\0, \: -1 \:and \: \frac{-3}{2}$

$Let \: \alpha = 0, \: \beta = -1 \:and \\\gamma = \frac{-3}{2}$

Verification:

$i) \alpha + \beta + \gamma \\= 0+(-1)+\frac{-3}{2}\\= -1-\frac{3}{2}\\= \frac{-2-3}{2}\\= \frac{-5}{2} \\=\frac{-b}{a}$

$ii )\alpha \beta + \beta \gamma + \gamma \alpha \\= 0\times (-1) + (-1) \times \frac{-3}{2} + \frac{-3}{2} \times 0 \\= \frac{3}{2} \\=\frac{c}{a}$

$iii) \alpha \beta \gamma \\= 0\times (-1)\times \frac{-3}{2}\\= 0 \\= \frac{-d}{a}$

•••♪