Use the following example to answer the question. Antoine goes to the mall to buy new shoes and jerseys. He pays a set rate per pair of shoes and a set rate per jersey, and he pays more per jersey than he pays per pair of shoes. His total spending can be modeled by the equation 35x+55y . Which of the statements correctly interprets a part of the expression? Statement #1: Antoine spends $35.00 per pair of shoes. Statement #2: Antoine spends $55.00 per pair of shoes. Statement #3: Antoine spends $35.00 in total for shoes. (1 point) Statement # correctly interprets a part of the expression.
None of the statements correctly interprets a part of the expression. The correct interpretation is that Antoine spends $35.00 per pair of shoes and $55.00 per jersey, not per pair of shoes. Thus, statement #1 is incorrect and statement #2 is incorrect. Statement #3 is also incorrect because it states that Antoine spends $35.00 in total for shoes, but that is not accurate.
Consider the expression (16−12)2+5(8+22)53−(1+4)2 . What is the simplified numerator?(1 point) Responses 54 54 42+60 4 squared plus 60 76 76 100 100
To simplify the numerator of the expression (16−12)2 + 5(8+22), we perform the operations inside the parentheses first: (16−12) = 4 and (8+22) = 30.
Plugging these values back into the expression, we have:
(4)^2 + 5(30)
Simplifying further, we have:
16 + 5(30)
Next, we evaluate the multiplication:
16 + 150
Finally, adding the two terms together, we get:
166
Therefore, the simplified numerator is 166.
Which expression correctly demonstrates the Commutative Property of Multiplication?(1 point) Responses 15⋅(x)⋅4=4⋅15⋅(x) 15 times left parenthesis x right parenthesis times 4 equals 4 times 15 times left parenthesis x right parenthesis 15⋅x⋅4=4+15+x 15 times x times 4 equals 4 plus 15 plus x x+4+15=15+4+x x plus 4 plus 15 equals 15 plus 4 plus x 15(x+4)=15(x⋅4)
The expression that correctly demonstrates the Commutative Property of Multiplication is:
15⋅(x)⋅4=4⋅15⋅(x)
This expression shows that the order of multiplication does not affect the result. The numbers 15, x, and 4 can be rearranged and the product will remain the same.
Write an equivalent expression to (a⋅b)⋅c using the Associative Property of Multiplication.(1 point)
An equivalent expression to (a⋅b)⋅c using the Associative Property of Multiplication is:
a⋅(b⋅c)
Which expression is equivalent to −3x−5 ?(1 point) Responses −1+(x−4x)+6 negative 1 plus left parenthesis x minus 4 x right parenthesis plus 6 (x−7)+(2−4x) left parenthesis x minus 7 right parenthesis plus left parenthesis 2 minus 4 x right parenthesis (−4x−6)+(2+5x) left parenthesis negative 4 x minus 6 right parenthesis plus left parenthesis 2 plus 5 x right parenthesis −1+3x−(2x+6)
The expression that is equivalent to -3x-5 is:
-1+(x-4x)+6
This expression can be simplified to
-1+(-3x)+6
which simplifies further to
-1-3x+6
and finally to
5-3x
According to the Distributive Property, which expression is equivalent to 4x3+8x2+12x ?(1 point) Responses 4x(x2+2x+3) 4 x left parenthesis x squared plus 2 x plus 3 right parenthesis (2x2+4)(2x2+3) left parenthesis 2 x squared plus 4 right parenthesis left parenthesis 2 x squared plus 3 right parenthesis (2x2+4)(2x−3) left parenthesis 2 x squared plus 4 right parenthesis left parenthesis 2 x minus 3 right parenthesis 4x(4x+5x)
The expression that is equivalent to 4x^3 + 8x^2 + 12x according to the Distributive Property is:
4x(x^2 + 2x + 3)
This is because we can distribute the 4x to each term inside the parentheses.
Which of the following simplifies to the same expression as 4x2−3x2+5x−24 ?(1 point) Responses 4x(x−6) 4 x left parenthesis x minus 6 right parenthesis 2(x−12) 2 left parenthesis x minus 12 right parenthesis (x+8)(x−3) left parenthesis x plus 8 right parenthesis left parenthesis x minus 3 right parenthesis (x+6)(x−4)
The expression that simplifies to the same expression as 4x^2 − 3x^2 + 5x − 24 is:
(x + 8)(x - 3)
To simplify, we use the distributive property and multiply each term in the first parentheses by each term in the second parentheses:
x * x + x * (-3) + 8 * x + 8 * (-3)
This further simplifies to:
x^2 - 3x + 8x - 24
Combining like terms, we get:
x^2 + 5x - 24
Which is the same expression as 4x^2 − 3x^2 + 5x − 24.
Add the polynomial expressions 5z−2z2−4z+5 and 6z2+3−7z .(1 point) Responses 11z2+z−2 11 z squared plus z minus 2 9z2−11z+8 9 z squared minus 11 z plus 8 4z2−6z+8 4 z squared minus 6 z plus 8 4z2−z+8 4 z squared minus z plus 8
To add the polynomial expressions 5z−2z^2−4z+5 and 6z^2+3−7z, we combine like terms:
First, let's combine the z^2 terms: -2z^2 + 6z^2 = 4z^2
Next, let's combine the z terms: 5z - 4z - 7z = -6z
Lastly, let's combine the constant terms: 5 + 3 = 8
Putting it all together, the sum of the two polynomial expressions is:
4z^2 - 6z + 8
Which of the following correctly simplifies (9x2+17x+13)−(2x2+12x+11) ?(1 point) Responses 7x2+5x+2 7 x squared plus 5 x plus 2 7x2+29x+24 7 x squared plus 29 x plus 24 12x2+2 12 x squared plus 2 9x2+17x+13−2x2−12x−11
To correctly simplify (9x^2 + 17x + 13) - (2x^2 + 12x + 11), we perform the subtraction operation by distributing the negative sign:
(9x^2 + 17x + 13) - (2x^2 + 12x + 11)
= 9x^2 + 17x + 13 - 2x^2 - 12x - 11
Next, combine like terms:
= (9x^2 - 2x^2) + (17x - 12x) + (13 - 11)
= 7x^2 + 5x + 2
Therefore, the correct simplification is: 7x^2 + 5x + 2.
Multiply the polynomials (3x−1)(2x+4)
To multiply the polynomials (3x−1)(2x+4), we use the distributive property and multiply each term in the first polynomial by each term in the second polynomial:
(3x−1)(2x+4) = 3x * 2x + 3x * 4 + (-1) * 2x + (-1) * 4
Simplifying further and combining like terms, we get:
6x^2 + 12x - 2x - 4
Combine like terms once again:
6x^2 + 10x - 4
Therefore, the product of the polynomials (3x−1)(2x+4) is: 6x^2 + 10x - 4.
Which of the following responses shows that polynomials form a closed system under multiplication?(1 point) Responses (x)(12x) left parenthesis x right parenthesis left parenthesis Start Fraction 1 over 2 End Fraction x right parenthesis 3x24−1 Start Fraction 3 x squared over 4 minus 1 End Fraction 6⋅−3 6 times negative 3 4x3⋅5 4 x cubed times 5
The response that shows that polynomials form a closed system under multiplication is:
4x^3⋅5
This is because the product of two polynomials is also a polynomial. In this case, multiplying the polynomial 4x^3 by the constant 5 gives us the polynomial 20x^3, which is still a polynomial.
Using the Commutative Property, prove that these algebraic expressions are equivalent. In 1–2 sentences, explain your result. Expression 1: y⋅3x Expression 2: 3xy
Expression 1: y⋅3x
Expression 2: 3xy
Using the Commutative Property of Multiplication, we can rearrange the terms in Expression 1 to match the order in Expression 2, resulting in 3xy. Therefore, Expression 1 and Expression 2 are equivalent.
Use the Associative Property to demonstrate that Expression 1 is equivalent to Expression 2. Expression 1: 22r−13 Expression 2: 19r+(3r−2)−11
Expression 1: 22r−13
Expression 2: 19r+(3r−2)−11
Using the Associative Property of Addition, we can rearrange the terms in Expression 2 by grouping them differently: 19r + (3r - 2) - 11 = 22r - 13. Therefore, Expression 1 and Expression 2 are equivalent.
Use the Distributive Property to prove that these algebraic expressions are equivalent. Explain your answer in one sentence for each expression and show your work. Expression 1: (x−5)(x+3) Expression 2: 2x2−x2−2x−20+5
Expression 1: (x−5)(x+3)
Expression 2: 2x^2−x^2−2x−20+5
Using the Distributive Property, we multiply each term inside the parentheses in Expression 1 by (x-5):
(x-5)(x+3) = (x*(x+3)) + (-5*(x+3))
= (x*x) + (x*3) + (-5*x) + (-5*3)
= x^2 + 3x - 5x - 15
= x^2 - 2x - 15
Therefore, Expression 1: (x−5)(x+3), is equivalent to Expression 2: 2x^2−x^2−2x−20+5 as both simplify to x^2 - 2x - 15.
In 1–2 sentences, explain why the following operation is a closed system under subtraction. (9x2+17x+13)−(2x2+12x+11)
The operation of subtracting two polynomials, (9x^2 + 17x + 13) - (2x^2 + 12x + 11), is a closed system because the difference of two polynomials always results in another polynomial, ensuring that the subtraction operation stays within the set of polynomials.
Use the following multiplication problem to show how multiplying polynomials is a closed system. (x2−1)(−2x+5)
To show how multiplying polynomials is a closed system, let's multiply the given polynomials: (x^2 - 1)(-2x + 5).
Expanding the brackets using the distributive property:
(x^2 * -2x) + (x^2 * 5) + (-1 * -2x) + (-1 * 5)
Simplifying:
-2x^3 + 5x^2 + 2x - 5
The result -2x^3 + 5x^2 + 2x - 5 is still a polynomial, which demonstrates that multiplying polynomials is a closed system.