(3x-1).jpeg "(4x^5+x^3-7x^2+2)(3x-1)")
Multiplying Polynomials: A Step-by-Step Guide
This article will guide you through the process of multiplying the polynomials (4x^5 + x^3 - 7x^2 + 2) and (3x - 1).
Understanding the Problem
We are tasked with multiplying two polynomials. This can be achieved using 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.
Step-by-Step Solution
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Distribute the first term of the second polynomial (3x):
- Multiply 3x by each term in the first polynomial:
- (3x) * (4x^5) = 12x^6
- (3x) * (x^3) = 3x^4
- (3x) * (-7x^2) = -21x^3
- (3x) * (2) = 6x
- Multiply 3x by each term in the first polynomial:
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Distribute the second term of the second polynomial (-1):
- Multiply -1 by each term in the first polynomial:
- (-1) * (4x^5) = -4x^5
- (-1) * (x^3) = -x^3
- (-1) * (-7x^2) = 7x^2
- (-1) * (2) = -2
- Multiply -1 by each term in the first polynomial:
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Combine the results:
- Add the terms obtained from both distributions:
- 12x^6 + 3x^4 - 21x^3 + 6x - 4x^5 - x^3 + 7x^2 - 2
- Add the terms obtained from both distributions:
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Simplify by combining like terms:
- 12x^6 - 4x^5 + 3x^4 - 21x^3 - x^3 + 7x^2 + 6x - 2
- 12x^6 - 4x^5 + 3x^4 - 22x^3 + 7x^2 + 6x - 2
The Final Result
The product of the polynomials (4x^5 + x^3 - 7x^2 + 2) and (3x - 1) is 12x^6 - 4x^5 + 3x^4 - 22x^3 + 7x^2 + 6x - 2.
Key Points to Remember
- Distributive Property: This is the core principle for multiplying polynomials.
- Combining Like Terms: Simplifying the expression by combining terms with the same variable and exponent.
This step-by-step method can be used to multiply any two polynomials. It is important to be methodical and to pay attention to signs when performing the calculations.