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Division Algorithm for Polynomials

Example: Find all the zeroes of ${2x^4 – 3x^3 – 3x^2 + 6x – 2}$, if you know that two of its zeroes are √2 and −√2 .

Solution : Since two zeroes are √2 and −√2, ${(x −√2 )(x +√2) = x^2 – √2^2}$ is a factor of the given polynomial.

Now, we divide the given polynomial by ${x^2 – 2}$.

${2x^4 – 3x^3 – 3x^2 + 6x – 2}$

${x^2 – 2}$

$2x^2$

$-3x$

$+1$

$2x^4 - 4x^2$

($-$)

($+$)

$-3x^3+x^2+6x - 2$

$-3x^3+6x$

($+$)

   ($-$)

$x^2-2$

$(-) (+)$

So, ${2x^4 – 3x^3 – 3x^2 + 6x – 2 = (x^2 – 2)(2x^2 – 3x + 1)}$.

Now, by splitting ${–3x}$, we factorise ${2x^2 – 3x + 1}$ as ${(2x – 1)(x – 1)}$. So, its zeroes are given by ${x=1/2}$ and ${x = 1}$.

Therefore, the zeroes of the given polynomial are √2,−√2, ${1/2}$,1.

Example: Find all the zeroes of the polynomial ${x^4-7x^2+10}$, if two of its zeroes are √5 and −√5 .

Solution : Since two zeroes are √5 and −√5, ${(x −√5 )(x +√5) = x^2 – √5^2}$ is a factor of the given polynomial.

Now, we divide the given polynomial by ${x^2 – 5}$.

Now divide ${x^4-7x^2+10}$ by ${x^2 – 5}$

${x^4-7x^2+10}$

${x^2 – 5}$

$x^2$

$-$

$2$

$x^4 - 5x^2$

($-$)

($+$)

$-2x^2+10$

($+$)

   ($-$)

$0 0$

Therefore ${x^4-7x^2+10}$ = $(x^2-5)(x^2-2)$
= $(x-√5)(x+√5)(x-√2)(x+√2)$

Ans:Hence, the zeroes of the polynomial are $√5, -√5, √2, -√2$

Example:Find all the zeroes of the polynomial ${x^4-3x^3-19x^2+27x+90}$, if two of its zeroes are $3$ and $−3$ .
Solution: Since $3$ and $-3$ are zeores of the polynomial, so $(x-3)(x+3)$ is a factor. So divide the polynomial ${x^4-3x^3-19x^2+27x+90}$ by $x^2-9$

${x^4-3x^3-19x^2+27x+90}$

$x^2-9$

$x^2$

$-3x$

$-10$

$x^4$       $-9x^2$

($-$)

     ($+$)

$-3x^3- 10 x^2+ 27x+90$

$-3x^3$       $+27x$

($+$)

      ($-$)

$-10x^2$      $+90$

$-10x^2$      $+90$

$(+)$         $(-)$

Since the reminder is zero, so $x^2-3x-10$ is also a factor of the polynomial. $x^2-3x-10$ => $x^2-5x+2x-10$ $x^2-3x-10$ => $x(x-5)+2(x-5)$ $x^2-3x-10$ => $(x-5)(x+2)$ So, the remaining roots of the polynomial are $x= 5$ and $-2$

Chapters For Class X- CBSE