Question

if the two factors of 2x^3+ax^2+bx+9 are x+3 and x-3 what are the values of a and b?

Answer 1

Answer:

a = -1

b = -18

Note:

★ The possible values of the variable for which the polynomial becomes zero are called its zeros .

★ If x - a is a factor of the given polynomial p(x) , then x = a is a zero of p(x) and hence p(a) = 0.

Solution:

Here,

The given polynomial is ;

2x² + ax² + bx + 9 .

Also,

It is given that , x + 3 and x - 3 are the factors of the polynomial .

If x + 3 = 0 , then

x = -3

If x - 3 = 0 , then

x = 3

Thus,

At x = - 3 , 3 the polynomial becomes zero .

This,

=> 2(-3)³ + a(-3)² + b(-3) + 9 = 0

=> -54 + 9a - 3b + 9 = 0

=> -45 + 9a - 3b = 0

=> 3(-15 + 3a - b) = 0

=> -15 + 3a - b = 0

=> 3a - b = 15 -----------(1)

Also,

=> 2(3)³ + a(3)² + b(3) + 9 = 0

=> 54 + 9a + 3b + 9 = 0

=> 63 + 9a + 3b = 0

=> 3(21 + 3a + b) = 0

=> 21 + 3a + b = 0

=> 3a + b = -21 ------------(2)

Now,

Adding eq-(1) and (2) , we get ;

=> 3a - b + 3a + b = 15 + (-21)

=> 3a + 3a = 15 - 21

=> 6a = -6

=> a = -6/6

=> a = -1

Now,

Putting a = -1 in eq-(2) , we get ;

=> 3a + b = -21

=> 3(-1) + b = -21

=> -3 + b = -21

=> b = -21 + 3

=> b = -18

Hence,

a = -1

b = -18