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

Jasonbradley

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Jun 16, 2024

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

This article will explore the process of multiplying the two polynomials: (3x^5-x^4-2x^3+x^2+4x+5) and (x^2-2x+2).

The Distributive Property

The key to multiplying polynomials is the distributive property. This states that to multiply a sum by a number, we multiply each term of the sum by the number.

In our case, we need to distribute each term of the first polynomial (3x^5-x^4-2x^3+x^2+4x+5) by each term of the second polynomial (x^2-2x+2). This can be visualized as follows:

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

Expanding and Combining Like Terms

Now we need to distribute each term and simplify:

• 3x^5(x^2 - 2x + 2): This expands to 3x^7 - 6x^6 + 6x^5.
• -x^4(x^2 - 2x + 2): This expands to -x^6 + 2x^5 - 2x^4.
• -2x^3(x^2 - 2x + 2): This expands to -2x^5 + 4x^4 - 4x^3.
• x^2(x^2 - 2x + 2): This expands to x^4 - 2x^3 + 2x^2.
• 4x(x^2 - 2x + 2): This expands to 4x^3 - 8x^2 + 8x.
• 5(x^2 - 2x + 2): This expands to 5x^2 - 10x + 10.

Finally, we combine like terms:

3x^7 - 6x^6 + 6x^5 - x^6 + 2x^5 - 2x^4 - 2x^5 + 4x^4 - 4x^3 + x^4 - 2x^3 + 2x^2 + 4x^3 - 8x^2 + 8x + 5x^2 - 10x + 10
= **3x^7 - 7x^6 + 6x^5 + 3x^4 - 6x^3 - x^2 - 2x + 10**

The Final Result

Therefore, the product of the two polynomials (3x^5-x^4-2x^3+x^2+4x+5) and (x^2-2x+2) is 3x^7 - 7x^6 + 6x^5 + 3x^4 - 6x^3 - x^2 - 2x + 10.