To demonstrate the Closure Property of Multiplication of polynomials using the given expression \((\frac{1}{2}x^2 - 3)(4y^3 + 5x^2)\), we will multiply these two polynomials together.
Let's break it down step-by-step:
-
Distribute each term in the first polynomial by each term in the second polynomial:
\[ \left(\frac{1}{2}x^2 - 3\right)(4y^3 + 5x^2) = \left(\frac{1}{2}x^2\right)(4y^3) + \left(\frac{1}{2}x^2\right)(5x^2) - 3(4y^3) - 3(5x^2) \]
-
Perform the multiplications:
- \(\left(\frac{1}{2}x^2\right)(4y^3) = \frac{4}{2}x^2y^3 = 2x^2y^3\)
- \(\left(\frac{1}{2}x^2\right)(5x^2) = \frac{5}{2}x^{2+2} = \frac{5}{2}x^4\)
- \(-3(4y^3) = -12y^3\)
- \(-3(5x^2) = -15x^2\)
-
Combine all the terms:
\[ 2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2 \]
-
Reorganize and combine like terms, if applicable:
The multiplication results in: \[ \frac{5}{2}x^4 + (2x^2y^3 - 15x^2 - 12y^3) \]
While there are no like terms to combine further (as we're dealing with different degrees and variables), we can express the final result concisely as:
\[ \frac{5}{2}x^4 + 2x^2y^3 - 15x^2 - 12y^3 \]
Conclusion:
The product of the two polynomials is a polynomial expressed by the equation:
\[ \frac{5}{2}x^4 + 2x^2y^3 - 15x^2 - 12y^3 \]
This demonstrates the Closure Property of Multiplication of polynomials, as the result is also a polynomial.
1 answer