Question

Multiply the polynomial (b+8)(3b−6) to simplify.

Answer 1

What is the product of the polynomials (x2y+2)(x2−y)?

Answer 2

Which expression is equivalent to x3(2+y5)?

Answer 3

Which of the following responses shows that polynomials form a closed system under multiplication?

Answer 4

Use multiplication to demonstrate these polynomials form a closed system.

(12x2−3)(4y3+5x2)

Answer 5

To multiply the polynomial, we use the distributive property.

Starting with the first term in the first polynomial, "b", we multiply it by each term in the second polynomial:

b * 3b = 3b²
b * -6 = -6b

Next, we move to the second term in the first polynomial, "+8", and do the same multiplication:

8 * 3b = 24b
8 * -6 = -48

Putting it all together, the product of (b+8)(3b-6) is:

3b² - 6b + 24b - 48

Now, we combine like terms:

3b² + 18b - 48

So, (b+8)(3b-6) simplifies to 3b² + 18b - 48.

Answer 6

To find the product of the polynomials, we use the distributive property.

Starting with the first term in the first polynomial, "x^2y", we multiply it by each term in the second polynomial:

(x^2y) * (x^2) = x^(2+2)y = x^4y
(x^2y) * (-y) = -x^2y^2

Next, we move to the second term in the first polynomial, "+2", and do the same multiplication:

(2) * (x^2) = 2x^2
(2) * (-y) = -2y

Putting it all together, the product of (x^2y+2)(x^2 − y) is:

x^4y - x^2y^2 + 2x^2 - 2y

So, (x^2y+2)(x^2 − y) simplifies to x^4y - x^2y^2 + 2x^2 - 2y.

Answer 7

To find the expression that is equivalent to x3(2+y5), we need to distribute the x³ to each term inside the parentheses.

Distributing x³ to 2 gives us:
x³ * 2 = 2x³

Distributing x³ to y⁵ gives us:
x³ * y⁵ = x³y⁵

Putting it all together, the expression equivalent to x³(2+y⁵) is:

2x³ + x³y⁵

Answer 8

To show that polynomials form a closed system under multiplication, we need to demonstrate that the product of any two polynomials is also a polynomial.

Let's consider two polynomials, P(x) and Q(x). The product of these two polynomials is denoted as P(x) * Q(x).

To prove that polynomials form a closed system under multiplication, we need to show that the product of P(x) and Q(x) is still a polynomial, meaning it can be expressed in the form of a polynomial with finite terms.

For example, let's say we have:
P(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ
Q(x) = b₀ + b₁x + b₂x² + ... + bₘxᵐ

The product of P(x) and Q(x) would be:
P(x) * Q(x) = (a₀ + a₁x + a₂x² + ... + aₙxⁿ)(b₀ + b₁x + b₂x² + ... + bₘxᵐ)

When we multiply these two polynomials, all the terms will be combined and arranged according to the powers of x, resulting in a new polynomial. This new polynomial will also have a finite number of terms in the form of x raised to some power.

Therefore, since the product of any two polynomials is still a polynomial, we can conclude that polynomials form a closed system under multiplication.

Answer 9

To demonstrate that the polynomials (12x² - 3) and (4y³ + 5x²) form a closed system under multiplication, we need to multiply them together and show that the result is also a polynomial.

Using the distributive property, we multiply each term in the first polynomial by each term in the second polynomial:

(12x² - 3)(4y³ + 5x²)

= 12x² * 4y³ + 12x² * 5x² - 3 * 4y³ - 3 * 5x²

= 48x²y³ + 60x⁴ - 12y³ - 15x²

As we can see, the result is a polynomial with finite terms, and it can be expressed in the form of a polynomial. Therefore, the polynomials (12x² - 3) and (4y³ + 5x²) form a closed system under multiplication.