a^2 - 2ab - 15b^2
This is what I got:
(a-5b)(a+3b)
Correct!
x^4+2x^3-4x^2-5x-6 DIVIDED BY x^2+x-2
My answer:
x^2+x-3 remainder of -12\x^2+x-2
To divide the polynomial (x^4+2x^3-4x^2-5x-6) by the polynomial (x^2+x-2), we can use long division. Here's how you can do it:
1. Divide the leading term of the numerator (x^4) by the leading term of the denominator (x^2). This gives us x^2 as the quotient.
x^2
_______________________
x^2+x-2 | x^4 + 2x^3 - 4x^2 - 5x - 6
2. Multiply the entire denominator (x^2+x-2) by the calculated quotient (x^2), and subtract it from the numerator.
x^2+x-2
_______________________
x^2+x-2 | x^4 + 2x^3 - 4x^2 - 5x - 6
- (x^4 + x^3 - 2x^2)
= x^3 - 2x^2 - 5x - 6
3. Repeat the process by dividing the leading term of the new numerator (x^3) by the leading term of the denominator (x^2). This gives us x as the new quotient.
x^2 + x
_______________________
x^2+x-2 | x^4 + 2x^3 - 4x^2 - 5x - 6
- (x^4 + x^3 - 2x^2)
_______________________
-x^3 - 2x^2 - 5x
+ (x^3 + x^2 - 2x)
= -3x^2 - 3x - 6
4. Divide the leading term of the new numerator (-3x^2) by the leading term of the denominator (x^2). This gives us -3 as the new quotient.
x^2 + x - 3
_______________________
x^2+x-2 | x^4 + 2x^3 - 4x^2 - 5x - 6
- (x^4 + x^3 - 2x^2)
_______________________
-x^3 - 2x^2 - 5x
+ (x^3 + x^2 - 2x)
_______________________
-3x^2 - 3x - 6
5. Multiply the entire denominator (x^2+x-2) by the calculated quotient (-3), and subtract it from the new numerator.
x^2 + x - 3
_______________________
x^2+x-2 | x^4 + 2x^3 - 4x^2 - 5x - 6
- (x^4 + x^3 - 2x^2)
_______________________
-x^3 - 2x^2 - 5x
+ (x^3 + x^2 - 2x)
_______________________
-3x^2 - 3x - 6
+ (3x^2 + 3x - 6)
= 0
Since the remainder is 0, the division is exact. The quotient is x^2 + x - 3.