The Polynomial P(x)=x³-4x+a when divided by the polynomial (x-3) leaves reminder 5. What is the value of a ? *
Answers
Answer:
a=1
Step-by-step explanation:
Given ax^3 + 4x^2 + 3x - 4 and x^3 - 4x + a leave the same remainder when divided by x - 3.
Let p(x) = ax^3 + 4x^2 + 3x - 4 and g(x) = x^3 - 4x + a
By remainder theorem, if f(x) is divided by (x − a) then the remainder is f(a)
Here when p(x) and g(x) are divided by (x − 3) the remainders are p(3) and g(3) respectively.
Also given that p(3) = g(3) → (1)
Put x = 3 in both p(x) and g(x)
Hence equation (1) becomes,
a(3)^3 + 4(3)^2 + 3(3) - 4 = (3)^3 - 4(3) + a
⇒ 27a + 36 + 9 − 4 = 27 − 12 + a
⇒ 27a + 41 = 15 + a
⇒ 26a = 15 − 41 = − 26
∴ a = −1
Alternate Method :
According to remainder theorem, if f(x) is divided by (x-a)then remainderis f(a)
f(x) = ax³+4x²+3x-4
g(x)= x³-4x +a
f(3)=A(27)+4(9)+3(3)-4
27a+41
g(3)=27-4(3)+a
15+a
f(3)=G(3)
27a+41=15+a
26a=15-41
a=15-41/26
a=-26/26
a=-1
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Step-by-step explanation:
x³-4x+a=5
substitute 5 in the place of x and equate it to 5
5³-4×5+a=5
125-20+a=5
105+a=5
105-5=a
100=a