Question

given (x+2) is a factor of the polynomial f(x)= 3x^3 +ax^2+4x+b and leaves a remainder of -5 when divided by (x+3), find the value of a and b

Answer 1

$\textbf{Remainder theorem:}$

$\text{The remainer when P(x) is divided by (x-a) is P(a)}$

$\textbf{Factor theorem:}$

$\text{(x-a) is a factor of f(x) iff f(a) =0}$

$\textbf{Given:}$

$f(x)=3x^3 +ax^2+4x+b$

$\text{since (x+2) is a factor of f(x), }f(-2)=0$

$\implies\;3(-2)^3 +a(-2)^2+4(-2)+b=0$

$\implies\;-24+4a-8+b=0$

$\implies\;4a+b=32$......(1)

$\text{when f(x) is divided by (x+3) it leaves the remainder -5}$

$\text{By, remainder theorem}$

$f(-3)=-5$

$\implies\;3(-3)^3 +a(-3)^2+4(-3)+b=-5$

$\implies\;-81+9a-12+b=-5$

$\implies\;9a+b=93-5$

$\implies\;9a+b=88$......(2)

$\text{(1)-(2) gives }$

$-5a=-56$

$\bf\;a=\frac{56}{5}$

$(1)\implies\;4(\frac{56}{5})+b=32$

$\frac{224}{5}+b=32$

$b=32-\frac{224}{5}$

$b=\frac{160-224}{5}$

$\bf\;b=\frac{-64}{5}$

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If a quadratic equation of the form ax^2 + c when divided by x and (x + 1) leaves remainder 2 and 4 respectively, then the value of a^2 + c^2 is

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Answer 2

Answer:

Step-by-step explanation: