Name: Division of a Trinomial by a Trinomial
Uploaded: 2019-07-19
Duration: 3 min 36 s
Description: Can we divide a polynomial by another polynomial to get the quotient polynomial and the remainder polynomial? Let's explore this concept in this video.

Question

Verify that the numbers given alongside the cubic polynomials below are their zeroes. Also, verify the relationship between the zeroes and the coefficients.

$3 x^{3} - 10 x^{2} - 27 x + 10$ ; $5$ , $- 2$ , $\frac{1}{3}$

Accepted Answer

Let $3 x^{3} - 10 x^{2} - 27 x + 10$

Given, the zeroes of $p (x)$ are $5$ , $- 2$ , $\frac{1}{3}$

Let $α = 5$ , $β = - 2$ , $γ = \frac{1}{3}$

Any number $c$ is a zero of a polynomial $f (x)$ , if $f (c) = 0$ .

$p (5) = 3 {(5)}^{3} - 10 {(5)}^{2} - 27 (5) + 10$

$= 375 - 250 - 135 + 10$

$= 385 - 385$

$= 0$

$p (- 2) = 3 {(- 2)}^{3} - 10 {(- 2)}^{2} - 27 (- 2) + 10$

$= - 24 - 40 + 54 + 10$

$= - 64 + 64$

$= 0$

$p (\frac{1}{3}) = 3 {(\frac{1}{3})}^{3} - 10 {(\frac{1}{3})}^{2} - 27 (\frac{1}{3}) + 10$

$= \frac{1}{9} - \frac{10}{9} - 9 + 10$

$= - 10 + 10$

$= 0$

As $p (5) = 0$ , $p (- 2) = 0$ and $p (\frac{1}{3}) = 0$ , we conclude that $5$ , $- 2$ and $\frac{1}{3}$ are the zeroes of $p (x)$ .

For any cubic polynomial $f (x) = a x^{3} + b x^{2} + c x + d$ , the relations between the zeroes, say $α$ , $β$ , $γ$ and the coefficients, $a$ , $b$ , $c$ , $d$ are,

$α + β + γ = - \frac{b}{a}$ , $α β + β γ + α γ = \frac{c}{a}$ , $α β γ = - \frac{d}{a}$

With $α = 5$ , $β = - 2$ , $γ = \frac{1}{3}$ , and $a = 3$ , $b = - 10$ , $c = - 27$ , $d = 10$ for $p (x)$ , we have

$α + β + γ = 5 - 2 + \frac{1}{3} = \frac{10}{3} = - \frac{(- 10)}{3} = - \frac{b}{a}$

$α β + β γ + α γ = - 10 - \frac{2}{3} + \frac{5}{3} = - 9 = \frac{- 27}{3} = \frac{c}{a}$

$α β γ = (5) (- 2) (\frac{1}{3}) = - \frac{10}{3} = - \frac{d}{a}$

Hence, the relationship between the zeroes and the coefficients is verified.

Verify that the numbers given alongside the cubic polynomials below are their zeroes. Also, verify the relationship between the zeroes and the coefficients.

$3 x^{3} - 10 x^{2} - 27 x + 10$ ; $5$ , $- 2$ , $\frac{1}{3}$

Important Questions on Polynomials

Learn and Prepare for any exam you want !

Verify that the numbers given alongside the cubic polynomials below are their zeroes. Also, verify the relationship between the zeroes and the coefficients. 3x3-10x2-27x+10; 5, -2, 13

Important Questions on Polynomials

Learn and Prepare for any exam you want !

Verify that the numbers given alongside the cubic polynomials below are their zeroes. Also, verify the relationship between the zeroes and the coefficients.

$3 x^{3} - 10 x^{2} - 27 x + 10$ ; $5$ , $- 2$ , $\frac{1}{3}$