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Here is a question from NCERT-Exampler pg-15 question no. 6

For what value of $a$ and $b$, are the zeros of $q(x)=x^3+2x^2+a$ also the zeros of polynomial $px(x)=x^5-x^4-4x^3+3x^2+3x+b$ ? Which zeros of $p(x)$ are not the zeros of $q(x)$?

this querstion's answer is

$a=-1, b=-2$; 1 and 2 are the zeros of $q(x)$but not the zeros of $p(x)$

now how to solve this question...

please help me to short out with this problem and it would be so kind if you show me steps how you have done this and please take this very urgent because tomarrow is my test...

thanks for your answers.

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$\displaystyle x^5-x^4-4x^3+3x^2+3x+b=(x^3+2x^2+a)(x^2-3x)+(3-a)x^2+(3+3a)x+b$ If $y$ is one of the common roots, $\displaystyle y^5-y^4-4y^3+3y^2+3y+b=(y^3+2y^2+a)(y^2-3y)+(3-a)y^2+(3+3a)y+b$ $\displaystyle\implies (3-a)y^2+(3+3a)y+b=0$ and $y^3+2y^2+a=0$ –  lab bhattacharjee May 12 at 15:00

1 Answer 1

All the zeros of $q$ are also among the zeros of $p$ if and only if $p(x)=r(x)q(x)$. So you want to do a long division of the cubic into the quintic, and see what the conditions on $a$ and $b$ are for the remainder to be zero.

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couldn't get the answer –  anni May 12 at 15:27
    
@anni Note that the claim is true only if one accounts for multiplicities of roots, else e.g. all the zeros of $\,x^2\,$ are among those of $\,x,\,$ but $\ x \ne x^2 r(x)\,$ for a polynomial $\,r(x)\in \Bbb C[x].\, $ But the idea of using the polynomial Division Algorithm and Remainder Theorem does work here. Where are you stuck? –  Bill Dubuque May 12 at 15:36
    
please tell me under my level –  anni May 12 at 15:40
    
@anni What is your level? Do you know the Euclidean Division Algorithm (with remainder) for polynomials? –  Bill Dubuque May 12 at 15:46
    
i am in 10th and as far you have asked me about that is : dividend=divisor * quotient + remainder –  anni May 12 at 15:54

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