-(5%2Bx%5E2-3x).jpeg "(x^5-x^2-3x^4+3x+5x^3-5) (5+x^2-3x)")
Multiplying Polynomials: (x^5-x^2-3x^4+3x+5x^3-5) (5+x^2-3x)
This article will walk you through the process of multiplying two polynomials: (x^5-x^2-3x^4+3x+5x^3-5) and (5+x^2-3x).
Understanding the Process
Multiplying polynomials involves distributing each term of one polynomial to every term of the other polynomial. This can be done using the FOIL method (First, Outer, Inner, Last) for binomials, or by using the distributive property for any number of terms.
Steps to Multiply the Polynomials
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Rearrange the terms: Start by rearranging both polynomials in descending order of their exponents. This makes the multiplication process more organized.
(x^5-3x^4+5x^3-x^2+3x-5) (x^2-3x+5)
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Distribute: Multiply each term of the first polynomial by each term of the second polynomial.
- x^5(x^2-3x+5) = x^7 - 3x^6 + 5x^5
- -3x^4(x^2-3x+5) = -3x^6 + 9x^5 - 15x^4
- 5x^3(x^2-3x+5) = 5x^5 - 15x^4 + 25x^3
- -x^2(x^2-3x+5) = -x^4 + 3x^3 - 5x^2
- 3x(x^2-3x+5) = 3x^3 - 9x^2 + 15x
- -5(x^2-3x+5) = -5x^2 + 15x - 25
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Combine like terms: After distributing, combine all the terms with the same exponent.
x^7 - 6x^6 + 19x^5 - 21x^4 + 28x^3 - 15x^2 + 30x - 25
Result
The product of the two polynomials (x^5-x^2-3x^4+3x+5x^3-5) and (5+x^2-3x) is: x^7 - 6x^6 + 19x^5 - 21x^4 + 28x^3 - 15x^2 + 30x - 25.
Conclusion
This process can be applied to any pair of polynomials, no matter how many terms they have. The key is to carefully distribute each term and then combine like terms to reach the final result. Remember to always keep the polynomial organized and use the distributive property effectively.