Question

if the polynomials 6x4+8x3+17x2+21x+7 is divide by another polynomial 3x2+4x+1 the rernocinder comes out to be ax+b, find a and b

Answer 1

Let P(x) = 6x⁴ + 8x³ + 17x² + 21x + 7 is divided by 3x² + 4x + 1 and remainder comes out ax + b .
first find factors of 3x² + 4x + 1
e.g., 3x² + 3x + x + 1
= 3x(x + 1) + 1(x +1)
= (3x + 1)(x + 1) , it means there are two zeros of it is x = -1/3 and -1
Now, if we put -1/3 and -1 in P(x) we get ax + b

So, P(-1/3) = 6(-1/3)⁴ + 8(-1/3)³ + 17(-1/3)² + 21(-1/3) + 7 = a × -1/3 + b
⇒2/27 - 8/27 + 17/9 - 7 + 7 = -a/3 + b
⇒ (2 - 8 + 51)/27 = -a/3 + b
⇒ 45/27 = -a/3 + b
⇒ -a + 3b = 5 -------(1)

Again, P(-1) = 6(-1)⁴ + 8(-1)³ + 17(-1)² + 21(-1) + 7 = a(-1) + b
⇒ 6 - 8 + 17 -21 + 7 = -a + b
⇒30 - 29 = -a + b
⇒1 = -a + b -------(2)

Solve equations (1) and (2)
a = 1 and b = 2

Answer 2

Solution :-

On dividing 6x⁴ + 8x³ + 17x² + 21x + 7 by 3x² + 4x + 1, we get.

                    _______________________
 3x² + 4x + 1) 6x⁴ + 8x³ + 17x² + 21x + 7 ( 2x² + 5

                      6x⁴ + 8x³ + 2x²
                    _      _       _
                    ______________________ 
                                        15x² + 21x + 7
                                       
                                        15x² + 20x + 5
                                      _        _        _            
                                      _____________
                                                       x + 2
                                      _____________                        

So, the remainder is x + 2

According to the given question.

ax + b = x + 2

Equating the coefficients from both the sides, we have -

⇒ ax = x  and  b = 2

Hence, on comparing it with ax + b, we get

a = 1 and b = 2

Answer.