-(x%5E2-3).jpeg "(2x^4-5x^2+x^3-3-3x) (x^2-3)")
Multiplying Polynomials: (2x^4-5x^2+x^3-3-3x) (x^2-3)
This article will guide you through the process of multiplying the two polynomials: (2x^4-5x^2+x^3-3-3x) and (x^2-3). We will use the distributive property to achieve this.
Understanding the Distributive Property
The distributive property states that multiplying a sum by a number is the same as multiplying each addend in the sum by the number and then adding the results. In the context of polynomials, we can extend this property to multiply two polynomials by distributing each term in the first polynomial to every term in the second polynomial.
Applying the Distributive Property
Let's break down the multiplication step by step:
-
Distribute the first term of the first polynomial (2x^4) to the second polynomial: (2x^4) * (x^2-3) = 2x^6 - 6x^4
-
Distribute the second term of the first polynomial (-5x^2) to the second polynomial: (-5x^2) * (x^2-3) = -5x^4 + 15x^2
-
Distribute the third term of the first polynomial (x^3) to the second polynomial: (x^3) * (x^2-3) = x^5 - 3x^3
-
Distribute the fourth term of the first polynomial (-3) to the second polynomial: (-3) * (x^2-3) = -3x^2 + 9
-
Distribute the fifth term of the first polynomial (-3x) to the second polynomial: (-3x) * (x^2-3) = -3x^3 + 9x
Combining Like Terms
Now that we've distributed all the terms, we need to combine any like terms:
2x^6 - 6x^4 - 5x^4 + 15x^2 + x^5 - 3x^3 - 3x^2 + 9 - 3x^3 + 9x
Combining the terms:
2x^6 + x^5 - 11x^4 - 6x^3 + 12x^2 + 9x + 9
Final Result
Therefore, the product of (2x^4-5x^2+x^3-3-3x) and (x^2-3) is 2x^6 + x^5 - 11x^4 - 6x^3 + 12x^2 + 9x + 9.