Question

8. The ratio of the remainders when x2 + bx + c is
divided by (x - 3) and (x - 2) respectively is 4:5,
then (7b + c) equals
(1) 30
(2) 26
127 -25
(4) -29​

Answer 1

Answer:

i think (4) -29 is correct answer

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Answer 2

Answer:

(4) -29

Step-by-step explanation:

Given:

The polynomial is, $p(x)=x^{2}+bx+c$

Ratio of remainders = 4:5

According to Remainder theorem, if a polynomial $p(x)$ is divided by a binomial $(x-a)$, then the remainder is $f(a)$.

Therefore, the remainders when $p(x)=x^{2}+bx+c$ is divided by binomials $(x-3)\textrm{ and }(x-2)$ are $f(3)\textrm{ and }f(2)$ respectively.

$f(3)=(3)^{2}+3b+c=9+3b+c\\f(2)=(2)^{2}+2b+c=4+2b+c$

Now, as per question, ratio of remainders is 4:5. So,

$\frac{f(3)}{f(2)}=\frac{4}{5}\\\\\frac{9+3b+c}{4+2b+c}=\frac{4}{5}\\\\5(9+3b+c)=4(4+2b+c)\\45+15b+5c=16+8b+4c\\(15b-8b)+(5c-4c)=16-45\\7b+c=-29$

Therefore, $7b+c=-29$.