Question

If the polynomials az + 4z2 + 3z - 4 and z - 4z + a leave the same remainder when

divided by z -3, find the value of a.

Answer 1

Question:-

If the polynomials az + 4z2 + 3z - 4 and z - 4z + a leave the same remainder when

divided by z -3, find the value of a.

Solution:-

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.

Answer 2

$\pink{\bold{\underline{\underline{Answer:-}}}}$

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.