-(x%5E2-1).jpeg "(2x^5-5x^3+x^2+3x-1) (x^2-1)")
Expanding the Expression (2x^5-5x^3+x^2+3x-1)(x^2-1)
This article will demonstrate the steps involved in expanding the expression (2x^5-5x^3+x^2+3x-1)(x^2-1).
Understanding the Problem
We have a product of two polynomial expressions:
- (2x^5-5x^3+x^2+3x-1) - A polynomial of degree 5
- (x^2-1) - A polynomial of degree 2
Our goal is to multiply these two expressions to obtain a single polynomial.
Expanding Using the Distributive Property
The distributive property states that a(b+c) = ab + ac. We can apply this property to our problem by considering the first polynomial as a single term:
(2x^5-5x^3+x^2+3x-1)(x^2-1) = (2x^5-5x^3+x^2+3x-1) * x^2 + (2x^5-5x^3+x^2+3x-1) * -1
Now, we distribute the x^2 and -1 terms:
= 2x^7 - 5x^5 + x^4 + 3x^3 - x^2 - 2x^5 + 5x^3 - x^2 - 3x + 1
Combining Like Terms
Finally, we combine the terms with the same powers of x:
= 2x^7 - 7x^5 + x^4 + 8x^3 - 2x^2 - 3x + 1
Conclusion
Therefore, the expanded form of (2x^5-5x^3+x^2+3x-1)(x^2-1) is 2x^7 - 7x^5 + x^4 + 8x^3 - 2x^2 - 3x + 1. This is a polynomial of degree 7.