SOLUTION: Please can you help me Find the values of a and b if ax^4+bx^3-8x^2+6x-6 has a remainder of 2x+1 when divided by x^2-1

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Please can you help me Find the values of a and b if ax^4+bx^3-8x^2+6x-6 has a remainder of 2x+1 when divided by x^2-1      Log On


   

Question 1186024: Please can you help me
Find the values of a and b if ax^4+bx^3-8x^2+6x-6 has a remainder of 2x+1 when divided by x^2-1
Found 2 solutions by MathLover1, Edwin McCravy:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!


equal given factor to zero
x%5E2-1=0
%28x-1%29%28x%2B1%29=0
=> x=1 or+x=-1
substitute the value of the factor as x+=+1 in the expression and equate to zero plus reminder as is a factor of the expression.

x+=+1
a%2A1%5E4%2Bb%2A1%5E3-8%2A1%5E2%2B6%2A1-6+=0%2B2%2A1%2B1+
a+%2B+b+-+8+=+3
a%2Bb+=3%2B8
a%2Bb+=11
a=11-b....eq.1

x+=+-1
a%2A%28-1%29%5E4%2Bb%2A%28-1%29%5E3-8%2A%28-1%29%5E2%2B6%2A%28-1%29-6+=0%2B2%2A%28-1%29%2B1+
a+-+b+-+20+=+-1
a+=+b+%2B20++-1
a=b%2B19....eq.2
from eq.1 and eq.2 we have
11-b=b%2B19
11-19=b%2Bb
2b=-8
b=-4
go to eq.1 and substitute b
a=11-%28-4%29....eq.1
a=11%2B4
a=15
check
%2815x%5E4-4x%5E3-8x%5E2%2B6x-6+%29%2F%28x%5E2-1%29



Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!

Here's another way to do it, by dividing directly with long division:

                    ax2+       bx+    (-8+a)
x2+0x-1)ax4+bx3-     8x2+      6x-         6
        ax4+0x3-     ax2
            bx3+(-8+a)x2+      6x
            bx3+     0x2-      bx
                (-8+a)x2+  (6+b)x-         6
                (-8+a)x2+      0x-    (-8+a)
                           (6+b)x+(-6+(-8+a))

For this remainder to be 2x+1,

6+b = 2        -6+(-8+a) = 1
  b = -4          -6-8+a = 1
                   -14+a = 1
                       a = 15
                       
Edwin