Question

4) when a polynomial 2x² + 3x tax tb is divided by (0-2)
leaves remainder a and (x+2) leaves remainder -
Find a and b​

Answer 1

Question :

When a polynomial 2x² + 3x + ax + b is divided by (x - 2) leaves remainder a and (x + 2) leaves remainder -b . Find a and b .

Answér :

a = -13/2 , b = -15/2

Note :

★ Remainder theorem : If a polynomial p(x) is divided by (x - c) , then the remainder obtained is given as R = p(c) .

★ Factor theorem :

• If the remainder obtained on dividing a polynomial p(x) by (x - c) is zero , ie. if R = p(c) = 0 , then (x - c) is a factor of the polynomial p(x) .
• If (x - c) is a factor of the polynomial p(x) , then the remainder obtained on dividing the polynomial p(x) by (x - c) is zero , ie. R = f(c) = 0 .

Solution :

Here ,

The given polynomial is 2x² + 3x + ax + b .

Let the given polynomial be p(x) .

Thus ,

p(x) = 2x² + 3x + ax + b

Now ,

According to the question , if the given polynomial p(x) is divided by (x - 2) then the remainder obtained is a .

Thus ,

=> R = a

=> p(2) = a

=> 2•2² + 3•2 + a•2 + b = a

=> 8 + 6 + 2a + b = a

=> 14 + 2a + b - a = 0

=> a + b + 14 = 0 -----------(1)

Also ,

If the given polynomial p(x) is divided by (x + 2) ie. {x - (-2)} then the remainder obtained is -b .

Thus ,

=> R = -b

=> p(-2) = -b

=> 2•(-2)² + 3•(-2) + a•(-2) + b = -b

=> 8 - 6 - 2a + b = -b

=> 2 - 2a + b + b = 0

=> 2 - 2a + 2b = 0

=> 2(1 - a + b) = 0

=> 1 - a + b = 0

=> a = 1 + b ---------(2)

Now ,

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

=> a + b + 14 = 0

=> 1 + b + b + 14 = 0

=> 2b + 15 = 0

=> 2b = -15

=> b = -15/2

Now ,

Using eq-(2) , we have ;

=> a = 1 + b

=> a = 1 - 15/2

=> a = (2 - 15)/2

=> a = -13/2

Hence ,

a = -13/2 , b = -15/2