Question

If the polynomials az3+4z2+3z-4and z3-4z+9leave the same remainder when divided by z-3. Find the value of a

Answer 1

Let f(z) = az³+4z²+3z-4

zero of the polynomial (z-3) = 3

(since,z-3=0⇒z=3)

So,replacing z by 3,

f(3) = a(3)³+4(3)²+3(3)-4

⇒f(3) = 27a+36+9-4

⇒f(3) = 27a+41

Let g(z) = z³-4z+a

zero of the polynomial (z-3) = 3

(since,z-3=0⇒z=3)

So,replacing z by 3,

g(3) = (3)³-4(3)+a

⇒g(3) = 27-12+a

⇒g(3) = 15+a

Given that the two polynomials leaves same remainder when divided by z-3

So,g(3)=f(3)

⇒15+a = 27a+41

⇒15-41 = 27a - a

⇒-26 = 26a

⇒a = -1.

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