(2x^2-5x^3+2x+2 X^4-1) (x^2-x-1)

4 min read Jun 16, 2024

Multiplying Polynomials: (2x^2-5x^3+2x+2 x^4-1) (x^2-x-1)

This article will guide you through the process of multiplying two polynomials: (2x^2-5x^3+2x+2 x^4-1) (x^2-x-1). We'll break down the steps and provide a clear explanation.

Understanding the Problem

We are asked to multiply two polynomials together. The first polynomial is a fourth-degree polynomial: 2x^4 - 5x^3 + 2x^2 + 2x - 1. The second polynomial is a second-degree polynomial: x^2 - x - 1.

The Distributive Property

The key to multiplying polynomials is the distributive property: we multiply each term of the first polynomial by each term of the second polynomial.

Step-by-Step Multiplication

Multiply the first term of the first polynomial by each term of the second polynomial.
```
(2x^4)(x^2) = 2x^6
(2x^4)(-x) = -2x^5
(2x^4)(-1) = -2x^4
```

Repeat step 1 for the second term of the first polynomial.

(-5x^3)(x^2) = -5x^5
(-5x^3)(-x) = 5x^4
(-5x^3)(-1) = 5x^3

Repeat step 1 for the remaining terms of the first polynomial.

(2x^2)(x^2) = 2x^4
(2x^2)(-x) = -2x^3
(2x^2)(-1) = -2x^2

(2x)(x^2) = 2x^3
(2x)(-x) = -2x^2
(2x)(-1) = -2x

(-1)(x^2) = -x^2
(-1)(-x) = x
(-1)(-1) = 1

Combine like terms.

2x^6 - 2x^5 - 5x^5 - 2x^4 + 5x^4 + 2x^4 + 5x^3 - 2x^3 + 2x^3 - 2x^2 - 2x^2 - x^2 + x - 2x + 1

Simplify the expression:

**2x^6 - 7x^5 + 5x^4 + 5x^3 - 5x^2 - x + 1**

The Result

The product of the two polynomials (2x^2-5x^3+2x+2 x^4-1) (x^2-x-1) is 2x^6 - 7x^5 + 5x^4 + 5x^3 - 5x^2 - x + 1.

Key Points

The distributive property is the key to multiplying polynomials.
Remember to combine like terms after multiplying each term.
The resulting polynomial will have a degree equal to the sum of the degrees of the original polynomials.

Let me know if you have any other polynomial multiplications you'd like to solve!