Question

$\sf\large\Green{Question:-}$
x³+ax²+bx+4 is divided by x-2, the remainder is 6. if it is divided by x+1, the remainder is -3. find a and b.​ Need answer in good content otherwise it will be reported .​

Answer 1

$\sf\small\underline\green{Given:-}$

$\rm{\implies P(x)=x^3+ax^2+bx+4}$

$\sf\small\underline\green{To\:Find:-}$

$\tt{\implies The\: value\:a\:and\:b=?}$

$\sf\small\underline\green{Solution:-}$

To calculate the value of a and b at first we have to find out the remainder with the help of given clue in question. Then setting up equation with the help of clue after that calculate the value of a and b.

$\sf\small\underline\green{Given\:in\:case-(i):-}$

$\rm{\longrightarrow P(x)=x^3+ax^2+bx+4}$

$\rm{\longrightarrow Divisor = x -2}$

$\rm{\longrightarrow Remainder = 6}$

• Now calculate remainder :-

$\rm{\longrightarrow x - 2 = 0 => x = 2}$

• Putting x = 2 in P(x) we get remainder :-]

$\rm{\longrightarrow P(x)=x^3+ax^2+bx+4}$

$\rm{\longrightarrow P(2) = (2)^3 +a(2)^2+b(2)+4}$

$\rm{\longrightarrow P(2)=8 + 4a + 2b + 4}$

$\rm{\longrightarrow Remainder = 4a + 2b + 12}$

$\rm{\longrightarrow 4a + 2b + 12 = 6}$

$\rm{\longrightarrow 4a + 2b = 6 - 12}$

$\rm\red{\longrightarrow 4a + 2b = -6------(i)}$

$\sf\small\underline\green{Given\:in\:case-(ii):-}$

$\tt{\longrightarrow P(x)=x^3+ax^2+bx+4}$

$\tt{\longrightarrow Divisor = x +1}$

$\tt{\longrightarrow Remainder = -3}$

• Now calculate remainder :-

$\tt{\longrightarrow x +1 = 0 => x = -1}$

• Putting x = -1 in P(x) we get remainder :-]

$\tt{\longrightarrow P(x)=x^3+ax^2+bx+4}$

$\tt{\longrightarrow P(-1) = (-1)^3 +a(-1)^2+b(-1)+4}$

$\tt{\longrightarrow P(-1)=-1 + a-b + 4}$

$\tt{\longrightarrow Remainder = a - b + 3}$

$\tt{\longrightarrow a-b+3 = -3}$

$\tt{\longrightarrow a-b = -3 - 3}$

$\tt\red{\longrightarrow a-b = -6------(ii)}$

• Now solving equation (i) and (ii) :-]

$\tt{\longrightarrow 4a + 2b = -6}$

$\tt{\longrightarrow 4a - 4b = -24}$

• By solving we get here :-]

$\tt{\longrightarrow 6b = 18 => b=3}$

• Putting the value of b = 3 in eq (ii):-]

$\rm{\longrightarrow a - b=-6}$

$\rm{\longrightarrow a - 3 = -6}$

$\rm{\longrightarrow a = -6 + 3}$

$\rm{\longrightarrow a = - 3}$

$\sf\large{Hence,}$

$\tt\blue{\:\:\:\:\:\:\:\implies The\: value\:of\:(a)=-3}$

$\tt\blue{\:\:\:\:\:\:\:\:\implies The\: value\:of\:(b)=3}$

Answer 2

See the attachments for solution

https://brainly.in/question/44546522?