Multiply the polynomial (b+8)(3b−6) to simplify.
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.
What is the product of the polynomials (x2y+2)(x2−y)?
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.
Which expression is equivalent to x3(2+y5)?
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
Which of the following responses shows that polynomials form a closed system under multiplication?
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.
Use multiplication to demonstrate these polynomials form a closed system.
(12x2−3)(4y3+5x2)
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.
To multiply the polynomials (b+8) and (3b-6), we can use the distributive property.
First, we distribute the first term of the first polynomial, which is b, to both terms of the second polynomial, 3b and -6:
b * 3b = 3b^2
b * (-6) = -6b
Next, we distribute the second term of the first polynomial, which is 8, to both terms of the second polynomial, 3b and -6:
8 * 3b = 24b
8 * (-6) = -48
Now, we combine like terms, which are terms that have the same variable and the same exponent:
3b^2 + (-6b) + 24b + (-48)
To simplify further, we can combine the middle terms (-6b and 24b) by adding their coefficients:
3b^2 + 18b + (-48)
Finally, we write the polynomial in descending order of exponents:
3b^2 + 18b - 48
Therefore, the simplified form of (b+8)(3b-6) is 3b^2 + 18b - 48.