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  • If the polynomials az^{3} + 4z^{2} + 3z – 4 and z^{3} – 4z + a leave the same remainder when divided by z – 3, find the value of a.

If the polynomials az3 + 4z2 + 3z – 4 and z3 – 4z + a leave the same remainder when divided by z – 3, find the value of a.

Answers (1)

a = –1 

Solution

Let       p(z)=az^{3}+4z^{2}+3z-4\\

q(z)=z^{3}-4z+a

According to remainder theorem when p(x)  is divided by (x+a)  then the remainder is (-a) .

So when p(z) is divided by z – 3 then remainder is given by p(3).      

p(3)=a \times 3^{3}+ 4 \times 3^{2}+3 \times 3-4\\p(3) =27a+36+9-4\\ p(3)=27a+41 \cdots \cdots (1)

            Similarly

q(3)= 3^{3}- 4 \times 3+a \\q(3) =27-12+a\\ q(3)=15+a \cdots \cdots (2)

According to question p(3) = q(3)
27a+41=15+a \\ 27a-a=15-41\\ 26a=-26\\ a=-\frac{26}{26}\\ a=-1

Hence the answer is a=-1.

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infoexpert24

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