Question
Use the use multiplication to find the product that demonstrates the closure property of multiplication of polynomials (one half x squared - 3)( 4 y cubed + 3x squared)
Answers
GPT-4o mini
To demonstrate the closure property of multiplication of polynomials, we will multiply the two given polynomials: \((\frac{1}{2} x^2 - 3)\) and \((4 y^3 + 3x^2)\).
We can do this by distributing each term in the first polynomial to each term in the second polynomial. Here's how that works step by step:
1. Distribute \(\frac{1}{2} x^2\) to each term in the second polynomial:
\[
\frac{1}{2} x^2 \cdot 4 y^3 = 2 x^2 y^3
\]
\[
\frac{1}{2} x^2 \cdot 3 x^2 = \frac{3}{2} x^4
\]
2. Distribute \(-3\) to each term in the second polynomial:
\[
-3 \cdot 4 y^3 = -12 y^3
\]
\[
-3 \cdot 3 x^2 = -9 x^2
\]
3. Combine all the results:
\[
2 x^2 y^3 + \frac{3}{2} x^4 - 12 y^3 - 9 x^2
\]
Now, let's collect like terms (though in this case, each term is unique):
So, the resulting polynomial is:
\[
\frac{3}{2} x^4 + 2 x^2 y^3 - 9 x^2 - 12 y^3
\]
Since the result is also a polynomial, we can conclude that the product demonstrates the closure property of multiplication of polynomials. This means that when you multiply two polynomials, the result is always another polynomial.
We can do this by distributing each term in the first polynomial to each term in the second polynomial. Here's how that works step by step:
1. Distribute \(\frac{1}{2} x^2\) to each term in the second polynomial:
\[
\frac{1}{2} x^2 \cdot 4 y^3 = 2 x^2 y^3
\]
\[
\frac{1}{2} x^2 \cdot 3 x^2 = \frac{3}{2} x^4
\]
2. Distribute \(-3\) to each term in the second polynomial:
\[
-3 \cdot 4 y^3 = -12 y^3
\]
\[
-3 \cdot 3 x^2 = -9 x^2
\]
3. Combine all the results:
\[
2 x^2 y^3 + \frac{3}{2} x^4 - 12 y^3 - 9 x^2
\]
Now, let's collect like terms (though in this case, each term is unique):
So, the resulting polynomial is:
\[
\frac{3}{2} x^4 + 2 x^2 y^3 - 9 x^2 - 12 y^3
\]
Since the result is also a polynomial, we can conclude that the product demonstrates the closure property of multiplication of polynomials. This means that when you multiply two polynomials, the result is always another polynomial.