-(x2-%2B-2---x).jpeg "(2x3 + X4 - 6x2 + 11x - 10) (x2 + 2 - X)")
Multiplying Polynomials: A Step-by-Step Guide
This article will walk you through the process of multiplying the following polynomials:
(2x³ + x⁴ - 6x² + 11x - 10) (x² + 2 - x)
We'll use the distributive property and a systematic approach to arrive at the final product.
Step 1: Organizing the terms
First, let's arrange both polynomials in descending order of their exponents:
(x⁴ + 2x³ - 6x² + 11x - 10) (x² - x + 2)
Step 2: Distributing each term
Now, we'll distribute each term of the first polynomial over the second polynomial. This means we'll multiply each term of the first polynomial by x², then by -x, and finally by 2.
1. Multiplying by x²:
- x⁴ * x² = x⁶
- 2x³ * x² = 2x⁵
- -6x² * x² = -6x⁴
- 11x * x² = 11x³
- -10 * x² = -10x²
2. Multiplying by -x:
- x⁴ * -x = -x⁵
- 2x³ * -x = -2x⁴
- -6x² * -x = 6x³
- 11x * -x = -11x²
- -10 * -x = 10x
3. Multiplying by 2:
- x⁴ * 2 = 2x⁴
- 2x³ * 2 = 4x³
- -6x² * 2 = -12x²
- 11x * 2 = 22x
- -10 * 2 = -20
Step 3: Combining like terms
We now have a list of terms. Let's combine the terms with the same exponents:
- x⁶
- 2x⁵ - x⁵ = x⁵
- -6x⁴ - 2x⁴ + 2x⁴ = -6x⁴
- 11x³ + 6x³ + 4x³ = 21x³
- -10x² - 11x² - 12x² = -33x²
- 10x + 22x = 32x
- -20
Step 4: The Final Result
Combining all the terms, the final product of the two polynomials is:
(x⁴ + 2x³ - 6x² + 11x - 10) (x² - x + 2) = x⁶ + x⁵ - 6x⁴ + 21x³ - 33x² + 32x - 20
This process might seem lengthy, but with practice, you'll find multiplying polynomials becomes much easier!