Question

Multiply the polynomial (b+8)(3b-6) to simplify
Answer
What is the product of the polynomials (x^2y+2)(x^2-y)?
Answer
Which expression is equivalent to x^3(2+y^5)?
A. 2x^3+x^3y^5
B. 2x^3+y^5
C. x^3+x^3+xy+xy+xy
D. x^3+2+y^5
Answer
Which of the following responses demonstrate that polynomials form a closed system under multiplication?
A. (x^2+1)(x-1/2)
B. (square root x)(x+1)
C. (x^2/2)(1/x)
D. x^2+2x+1
Answer
Use the multiplication to find the product that demonstrates the closure property of multiplication of polynomials
(1/2x^2-3)(4y^3+5x^2)
Answered by GPT-4o mini
To multiply the polynomials \((b + 8)(3b - 6)\), we will use the distributive property (also known as the FOIL method for binomials).

1. **First:** Multiply the first terms in each binomial:
\[
b \cdot 3b = 3b^2
\]

2. **Outer:** Multiply the outer terms:
\[
b \cdot (-6) = -6b
\]

3. **Inner:** Multiply the inner terms:
\[
8 \cdot 3b = 24b
\]

4. **Last:** Multiply the last terms in each binomial:
\[
8 \cdot (-6) = -48
\]

Now combine all the results:
\[
3b^2 - 6b + 24b - 48
\]

Combine like terms \(-6b + 24b\):
\[
3b^2 + 18b - 48
\]

Thus, the simplified result of multiplying the polynomials \((b + 8)(3b - 6)\) is:
\[
\boxed{3b^2 + 18b - 48}
\]
Answered by GPT-4o mini
To find the product of the polynomials \((x^2y + 2)(x^2 - y)\), we will distribute each term in the first polynomial by each term in the second polynomial.

1. **First, distribute \(x^2y\):**
\[
x^2y \cdot x^2 = x^4y
\]
\[
x^2y \cdot (-y) = -x^2y^2
\]

2. **Next, distribute \(2\):**
\[
2 \cdot x^2 = 2x^2
\]
\[
2 \cdot (-y) = -2y
\]

Now, combine all these results:
\[
x^4y - x^2y^2 + 2x^2 - 2y
\]

Thus, the product of the polynomials \((x^2y + 2)(x^2 - y)\) is:
\[
\boxed{x^4y - x^2y^2 + 2x^2 - 2y}
\]
Answered by GPT-4o mini
To simplify the expression \( x^3(2 + y^5) \), we can distribute \( x^3 \) across the terms inside the parentheses.

\[
x^3(2 + y^5) = x^3 \cdot 2 + x^3 \cdot y^5 = 2x^3 + x^3y^5
\]

Now, let's compare this result with the provided options:

A. \( 2x^3 + x^3y^5 \)
B. \( 2x^3 + y^5 \)
C. \( x^3 + x^3 + xy + xy + xy \)
D. \( x^3 + 2 + y^5 \)

The correct expression equivalent to \( x^3(2 + y^5) \) is **A. \( 2x^3 + x^3y^5 \)**.

So, the answer is:
\[
\boxed{A}
\]
Answered by GPT-4o mini
To determine which of the given responses demonstrates that polynomials form a closed system under multiplication, we need to identify which option results in a polynomial when you multiply the expressions together.

**A.** \((x^2 + 1)(x - \frac{1}{2})\)
- This is a product of two polynomials. The result will be a polynomial.

**B.** \((\sqrt{x})(x + 1)\)
- \(\sqrt{x}\) is not a polynomial since it contains a fractional exponent. The result will not be a polynomial.

**C.** \((\frac{x^2}{2})(\frac{1}{x})\)
- This can be simplified to \(\frac{x^2}{2} \cdot \frac{1}{x} = \frac{x}{2}\), which is a polynomial, but since you are multiplying a polynomial by a rational function (not a polynomial), this doesn't demonstrate closure of polynomials.

**D.** \(x^2 + 2x + 1\)
- This is already a polynomial, but it does not represent a multiplication operation involving two polynomials.

Thus, the only option that showcases the closure of polynomials under multiplication is:

**A.** \((x^2 + 1)(x - \frac{1}{2})\)

So, the answer is:
\[
\boxed{A}
\]
Answered by GPT-4o mini
To find the product of the polynomials \(\left(\frac{1}{2}x^2 - 3\right)(4y^3 + 5x^2)\), we will distribute each term in the first polynomial to each term in the second polynomial.

1. **Distribute \(\frac{1}{2}x^2\):**
- \(\frac{1}{2}x^2 \cdot 4y^3 = 2x^2y^3\)
- \(\frac{1}{2}x^2 \cdot 5x^2 = \frac{5}{2}x^4\)

2. **Distribute \(-3\):**
- \(-3 \cdot 4y^3 = -12y^3\)
- \(-3 \cdot 5x^2 = -15x^2\)

Now, combine all the results together:
\[
2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2
\]

We can rearrange it for clarity:
\[
\frac{5}{2}x^4 + 2x^2y^3 - 15x^2 - 12y^3
\]

Thus, the product of \(\left(\frac{1}{2}x^2 - 3\right)(4y^3 + 5x^2)\) is:
\[
\boxed{\frac{5}{2}x^4 + 2x^2y^3 - 15x^2 - 12y^3}
\]

This product is a polynomial, demonstrating the closure property of multiplication of polynomials.