-(1%2Bx%5E3).jpeg "(3x^2+3x^5+x^3-4x-4x^4+1) (1+x^3)")
Expanding the Polynomial Expression: (3x^2+3x^5+x^3-4x-4x^4+1) (1+x^3)
This article explores the expansion of the polynomial expression (3x^2+3x^5+x^3-4x-4x^4+1) (1+x^3).
Understanding Polynomial Multiplication
To expand this expression, we employ the distributive property of multiplication. This involves multiplying each term in the first polynomial by each term in the second polynomial.
Expanding the Expression
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Distributing the first term of the second polynomial:
- (3x^2+3x^5+x^3-4x-4x^4+1) * 1 = 3x^2+3x^5+x^3-4x-4x^4+1
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Distributing the second term of the second polynomial:
- (3x^2+3x^5+x^3-4x-4x^4+1) * x^3 = 3x^5 + 3x^8 + x^6 - 4x^4 - 4x^7 + x^3
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Combining like terms:
The expanded expression becomes: 3x^8 + 3x^7 + x^6 - 4x^5 - 4x^4 + 4x^3 - 4x + 1
Resulting Polynomial
The expanded form of the given expression is: 3x^8 + 3x^7 + x^6 - 4x^5 - 4x^4 + 4x^3 - 4x + 1
This is an 8th-degree polynomial.
Conclusion
Expanding polynomial expressions like this is a fundamental concept in algebra. It allows us to manipulate and simplify expressions, which is crucial for solving equations and analyzing functions. Understanding the distributive property and combining like terms are essential skills for effectively working with polynomials.