If the polynomials ax^3 + 4x^2 + 3x – 4 and x^3 – 4x + a leave the same remainder when divided by (x – 3), the value of a is : -Maths 9th

If the polynomials ax^3 + 4x^2 + 3x – 4 and x^3 – 4x + a leave the same remainder when divided by (x – 3), the value of a is : -Maths 9th
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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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