(x%5E4%2B3x%5E2-4).jpeg "(2x^5-5x^3)(x^4+3x^2-4)")
Multiplying Polynomials: (2x^5 - 5x^3)(x^4 + 3x^2 - 4)
This article will guide you through the process of multiplying the two polynomials: (2x^5 - 5x^3) and (x^4 + 3x^2 - 4). We will use the distributive property to simplify the expression.
Understanding the Distributive Property
The distributive property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. In simpler terms, we "distribute" the multiplication.
Applying the Distributive Property
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Multiply the first term of the first polynomial (2x^5) by each term in the second polynomial:
- (2x^5)(x^4) = 2x^9
- (2x^5)(3x^2) = 6x^7
- (2x^5)(-4) = -8x^5
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Multiply the second term of the first polynomial (-5x^3) by each term in the second polynomial:
- (-5x^3)(x^4) = -5x^7
- (-5x^3)(3x^2) = -15x^5
- (-5x^3)(-4) = 20x^3
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Combine all the results:
2x^9 + 6x^7 - 8x^5 - 5x^7 - 15x^5 + 20x^3
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Simplify by combining like terms:
2x^9 + x^7 - 23x^5 + 20x^3
Final Result
The product of (2x^5 - 5x^3) and (x^4 + 3x^2 - 4) is 2x^9 + x^7 - 23x^5 + 20x^3.
Key Takeaways
- The distributive property is essential for multiplying polynomials.
- Remember to multiply each term in the first polynomial by each term in the second polynomial.
- Combine like terms to simplify the final expression.
This process can be applied to multiply any two polynomials, regardless of their complexity.