-(2x%5E2-x%2B1).jpeg "(2x^3+5x^2-2x+3) (2x^2-x+1)")
Multiplying Polynomials: (2x^3 + 5x^2 - 2x + 3)(2x^2 - x + 1)
This article will guide you through the process of multiplying two polynomials: (2x^3 + 5x^2 - 2x + 3) and (2x^2 - x + 1).
Understanding the Process
Multiplying polynomials involves distributing each term of one polynomial with every term of the other polynomial. We can use the FOIL method (First, Outer, Inner, Last) or the distributive property to achieve this.
Using the Distributive Property
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Distribute the first term of the first polynomial (2x^3) to each term of the second polynomial: (2x^3)(2x^2) + (2x^3)(-x) + (2x^3)(1) = 4x^5 - 2x^4 + 2x^3
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Distribute the second term of the first polynomial (5x^2) to each term of the second polynomial: (5x^2)(2x^2) + (5x^2)(-x) + (5x^2)(1) = 10x^4 - 5x^3 + 5x^2
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Distribute the third term of the first polynomial (-2x) to each term of the second polynomial: (-2x)(2x^2) + (-2x)(-x) + (-2x)(1) = -4x^3 + 2x^2 - 2x
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Distribute the fourth term of the first polynomial (3) to each term of the second polynomial: (3)(2x^2) + (3)(-x) + (3)(1) = 6x^2 - 3x + 3
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Combine like terms to simplify the expression: 4x^5 - 2x^4 + 2x^3 + 10x^4 - 5x^3 + 5x^2 - 4x^3 + 2x^2 - 2x + 6x^2 - 3x + 3
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The final result is: 4x^5 + 8x^4 - 7x^3 + 13x^2 - 5x + 3
Using the FOIL Method
The FOIL method is a shortcut for multiplying binomials, but it can be applied to larger polynomials by breaking them into smaller parts.
- Multiply the First terms: (2x^3)(2x^2) = 4x^5
- Multiply the Outer terms: (2x^3)(1) = 2x^3
- Multiply the Inner terms: (3)(2x^2) = 6x^2
- Multiply the Last terms: (3)(1) = 3
- Combine these results, and then distribute the remaining terms: (4x^5 + 2x^3 + 6x^2 + 3) + (5x^2 - 2x) (2x^2 - x + 1)
- Continue using FOIL or distributive property to multiply the remaining terms, and then combine like terms to reach the final result: 4x^5 + 8x^4 - 7x^3 + 13x^2 - 5x + 3
Conclusion
Multiplying polynomials requires careful distribution and combining of like terms. Whether using the distributive property or the FOIL method, the result will be the same: 4x^5 + 8x^4 - 7x^3 + 13x^2 - 5x + 3.