When a polynomial 2x^3+3x^2+ax+b is divided by x-2 leaves remainder 2 and when divided by x+2 givea remainder -2 then find a and b
Answers
Answer:
a = -7
b = -12
Explanation:
When a polynomial 2x^3+3x^2+ax+b is divided by x-2 leaves remainder 2 and when divided by x+2 givea remainder -2 then find a and b
f(x) = 2x³ + 3x² + ax + b
dividing by x-2 leaves remainder 2
f(2) = 2
=> 2(2)³ + 3(2)² +2a + b = 2
=> 16 + 12 + 2a + b = 2
=> 2a + b = -26 Eq1
dividing by x+2 leaves remainder -2
=> f(-2) = -2
=> 2(-2)³ + 3(-2)² -2a + b = -2
=> -16 + 12 - 2a + b = -2
=> -2a + b = 2 Eq 2
Adding eq 1 & eq 2
=> 2b = -24
=> b = -12
=> 2a -12 = -26
=> 2a = -14
=> a = -7
Answer:
a=(-7) and b=(-12)
Explanation:
Given,
p(x)=2x^3+3x^2+ax+b
g(x)=x-2=0=>x=2 [as -2 coming to RIGHT HAND SIDE(R.H.S)]
f(x)=x+2=0=>x=(-2) [AS 2 IS COMING TO R.H.S]
now, substitute x value in p(x)......
p(2)=>2(2)^3+3(2)^2+a(2)+b=2{ we know the remainder we have to takep(x)=2}
=>16+12+2a+b=2
=>28+2a+b=2
=>2a+b=2-28 [as 28 comes to R.H.S, it becomes -28]
=>2a+b=-26---------{equation 1}
now take p(x)=(-2)
p(-2)=>2(-2)^3+3(-2)^2+a(-2)+b=(-2)
=>(-16)+12+(-2a)+b=(-2)
=> -4-2a+b=(-2)
=>-2a+b=(-2)+4
=>-2a+b=2-----------{equation 2}
now take equation 1 +equation 2
2a+b= -26
+(-2a)+b=2
2b=-24
--------------------
b=-24/2
b=-12
substitute value of b in equation 1
2a+(-12)= -26
2a-12= -26
2a= -26+12
2a= -14
a= -14/7
a=-7
I HOPE MY ANSWER HELPED YOU......................