The polynomial p(x)=x^{4}-2 x^{3}+3 x^{2}-a x | vedclass

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Question

We know that if $p(x)$ is divided by $x+a,$ then the remainder $=p(-a)$

Now, $p(x)=x^{4}-2 x^{3}+3 x^{3}-a x+3 a-7$ is divided by $x+1,$ then the r

Vedclass · Accepted Answer

$62$

Vedclass · Answer

We know that if $p(x)$ is divided by $x+a,$ then the remainder $=p(-a)$

Now, $p(x)=x^{4}-2 x^{3}+3 x^{3}-a x+3 a-7$ is divided by $x+1,$ then the remainder $=p(-1)$

Now, $p(-1)=(-1)^{4}-2(-1)^{3}+3(-1)^{2}-a(-1)+3 a-7$

$=1-2(-1)+3(1)+a+3 a-7$

$=1+2+3+4 a-7$

$=-1+4 a$

Also, remainder $=19$

$	herefore \quad-1+4 a=19$

$\Rightarrow \quad 4 a=20 ; a=20 \div 4=5$

Again, when $p ( x )$ is divided by $x+2,$ then

Remainder $=p(-2)=(-2)^{4}-2(-2)^{3}+3(-2)^{2}-a(-2)+3 a-7$

$=16+16+12+2 a+3 a-7$

$=37+5 a$

$=37+5(5)=37+25=62$

The polynomial $p(x)=x^{4}-2 x^{3}+3 x^{2}-a x+3 a-7$ when divided by $x+1$ leaves the remainder $19 .$ Find the values of $a .$ Also find the remainder when $p(x)$ is divided by $x+2.$

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