Multiply the polynomial (b+8)(3b−6) to simplify.(1 point) Responses 3b2+30b−48 3 b squared plus 30 b minus 48 3b2+18b−48 3 b squared plus 18 b minus 48 b2+30b−48 b squared plus 30 b minus 48 b2+18b−36
To multiply the polynomials (b+8)(3b-6), we can use the distributive property.
(b+8)(3b-6) = (b)(3b) + (b)(-6) + (8)(3b) + (8)(-6)
Simplifying this further:
= 3b² - 6b + 24b - 48
= 3b² + 18b - 48
Therefore, the simplified product of the polynomials is 3b² + 18b - 48.
What is the product of the polynomials (x2y+2)(x2−y)?(1 point) Responses x4y−x2y2+2x2−2y x superscript 4 baseline y minus x squared y squared plus 2 x squared minus 2 y x4y+x2−2y x superscript 4 baseline y plus x squared minus 2 y x3y−x2y2+2x2−2y x cubed y minus x squared y squared plus 2 x squared minus 2 y x4y−xy+x2y
To multiply the polynomials (x^2y+2)(x^2−y), we can use the distributive property.
(x^2y+2)(x^2−y) = (x^2y)(x^2) + (x^2y)(-y) + (2)(x^2) + (2)(-y)
Simplifying this further:
= x^4y - xy^2 + 2x^2 - 2y
Therefore, the product of the polynomials (x^2y+2)(x^2−y) is x^4y - xy^2 + 2x^2 - 2y.
Which expression is equivalent to x3(2+y5) ?(1 point) Responses x3+2+y5 x cubed plus 2 plus y superscript 5 baseline 2x3+x3y5 2 x cubed plus x cubed y superscript 5 baseline 2x3+y5 2 x cubed plus y superscript 5 baseline x3+x3+xy+xy+xy
To simplify the expression x^3(2+y^5), we apply the distributive property:
x^3(2+y^5) = (x^3)(2) + (x^3)(y^5)
Simplifying further:
= 2x^3 + x^3y^5
Therefore, the expression equivalent to x^3(2+y^5) is 2x^3 + x^3y^5.
Which of the following responses demonstrates that polynomials form a closed system under multiplication?(1 point) Responses (x−−√)(x+1) left parenthesis Start Root x End Root right parenthesis left parenthesis x plus 1 right parenthesis (x22)(1x) left parenthesis Start Fraction x squared over 2 End Fraction right parenthesis left parenthesis Start Fraction 1 over x End Fraction right parenthesis (x2+1)(x−12) left parenthesis x squared plus 1 right parenthesis left parenthesis x minus Start Fraction 1 over 2 End Fraction right parenthesis x2+2x+1
The response that demonstrates that polynomials form a closed system under multiplication is:
(x^2 + 1)(x - 1/2)
In this response, both factors are polynomials and the product is also a polynomial. This shows that the set of polynomials is closed under multiplication.
Use multiplication to find the product that demonstrates the Closure Property of multiplication of polynomials. (12x2−3)(4y3+5x2) (1 point) Responses 52x2y3+2x4−8y3+15x2 Start Fraction 5 over 2 End Fraction x squared y cubed plus 2 x superscript 4 baseline minus 8 y cubed plus 15 x squared x2−−√y3+4x4−15x2 Start Root x squared End Root y cubed plus 4 x superscript 4 baseline minus 15 x squared 2x2y3+52x4−12y3−15x2 2 x squared y cubed plus Start Fraction 5 over 2 End Fraction x superscript 4 baseline minus 12 y cubed minus 15 x squared x2y3+5x4−8y3−15x2 x squared y cubed plus Start Fraction 5 over x superscript 4 baseline End Fraction minus 8 y cubed minus 15 x squared
To find the product (12x^2−3)(4y^3+5x^2) and demonstrate the Closure Property of multiplication of polynomials, we'll use the distributive property.
(12x^2−3)(4y^3+5x^2) = (12x^2)(4y^3) + (12x^2)(5x^2) + (-3)(4y^3) + (-3)(5x^2)
Simplifying further:
= 48x^2y^3 + 60x^4 - 12y^3 - 15x^2
Therefore, the product (12x^2−3)(4y^3+5x^2) is 48x^2y^3 + 60x^4 - 12y^3 - 15x^2. This demonstrates the Closure Property of multiplication of polynomials.
To multiply the polynomial (b+8)(3b-6), you can use the distributive property.
Step 1: Multiply the first terms of each binomial.
(b)(3b) = 3b^2
Step 2: Multiply the outer terms of each binomial.
(b)(-6) = -6b
Step 3: Multiply the inner terms of each binomial.
(8)(3b) = 24b
Step 4: Multiply the last terms of each binomial.
(8)(-6) = -48
Step 5: Simplify the terms obtained in steps 2, 3, and 4 by combining like terms.
-6b + 24b = 18b
Putting it all together, the simplified expression is:
3b^2 + 18b - 48.
Therefore, the correct answer is 3b^2 + 18b - 48.