(4m%5E2%2Bm-5).jpeg "(-2m^3+3m^2-m)(4m^2+m-5)")
Multiplying Polynomials: (-2m^3 + 3m^2 - m)(4m^2 + m - 5)
This article will guide you through the process of multiplying the two polynomials: (-2m^3 + 3m^2 - m) and (4m^2 + m - 5).
Understanding the Process
Multiplying polynomials involves distributing each term of the first polynomial to every term of the second polynomial. This is often referred to as the FOIL method (First, Outer, Inner, Last) for binomials, but for larger polynomials, we need a more generalized approach.
Step-by-Step Solution
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Distribute the first term of the first polynomial:
- (-2m^3) * (4m^2 + m - 5) = -8m^5 - 2m^4 + 10m^3
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Distribute the second term of the first polynomial:
- (3m^2) * (4m^2 + m - 5) = 12m^4 + 3m^3 - 15m^2
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Distribute the third term of the first polynomial:
- (-m) * (4m^2 + m - 5) = -4m^3 - m^2 + 5m
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Combine like terms:
- -8m^5 - 2m^4 + 10m^3 + 12m^4 + 3m^3 - 15m^2 - 4m^3 - m^2 + 5m
- Simplifying: -8m^5 + 10m^4 + 9m^3 - 16m^2 + 5m
Final Result
Therefore, the product of the polynomials (-2m^3 + 3m^2 - m) and (4m^2 + m - 5) is -8m^5 + 10m^4 + 9m^3 - 16m^2 + 5m.
Key Points to Remember
- Distribution: The key to multiplying polynomials is to distribute each term of the first polynomial to every term of the second polynomial.
- Combine like terms: After distributing, always simplify the expression by combining like terms.
- Order of operations: Remember to follow the order of operations (PEMDAS/BODMAS) when simplifying the expression.