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Factor the expression completely.

45x−18y+36


(1 point)
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11 answers

To factor the expression \( 45x - 18y + 36 \) completely, we can start by looking for the greatest common factor (GCF) of the coefficients.

  1. The coefficients are 45, -18, and 36.
  2. The GCF of these numbers is 9.

Next, we will factor out 9 from the expression:

\[ 45x - 18y + 36 = 9(5x - 2y + 4) \]

Now we need to check if the expression inside the parentheses, \( 5x - 2y + 4 \), can be factored further.

Unfortunately, \( 5x - 2y + 4 \) does not factor nicely using integers or simple monomials, as it doesn't have any common factors or can be rewritten as a product of simpler polynomials.

So, the completely factored form of the expression \( 45x - 18y + 36 \) is:

\[ \boxed{9(5x - 2y + 4)} \]

Factor this expression completely.

−21x+28


(1 point)
Responses

−7(3x−4)
negative 7 times open paren 3 x minus 4 close paren

−7(3x+4)
negative 7 times open paren 3 x plus 4 close paren

7(3x−4)
7 times open paren 3 x minus 4 close paren

−(21x−4)

To factor the expression \(-21x + 28\), we start by finding the greatest common factor (GCF) of the terms.

  1. The coefficients are -21 and 28. The GCF is 7.
  2. We also note that the first term is negative, so we will factor out \(-7\).

Now let's factor out \(-7\):

\[ -21x + 28 = -7(3x - 4) \]

So the correct factored form of the expression \(-21x + 28\) is:

\[ \boxed{-7(3x - 4)} \]

This matches the first response option.

Which factorization is equivalent to this expression?

−42k−54


(1 point)
Responses

−6(7k−9)
negative 6 times open paren 7 k minus 9 close paren

6(−7k+9)
6 times open paren negative 7 k plus 9 close paren

−6(7k+9)
negative 6 times open paren 7 k plus 9 close paren

6(7k+9)
6 times open paren 7 k plus 9 close paren

To find the correct factorization of the expression \(-42k - 54\), we first look for the greatest common factor (GCF).

  1. The coefficients are -42 and -54. The GCF of these numbers is 6.
  2. Since the expression has a negative sign, we will factor out \(-6\).

Now let's factor out \(-6\):

\[ -42k - 54 = -6(7k + 9) \]

So the correct factorization of the expression \(-42k - 54\) is:

\[ \boxed{-6(7k + 9)} \]

None of the provided responses directly match this factorization, but the equivalent rewritten form can be expressed by factoring out both the common factor and the negative sign:

  • Response Option: \(-6(7k + 9)\) which is equivalent to \(-6(7k + 9)\) but is not listed directly.

If you need an answer in those exact options, the closest one that represents the same value would be:

\[ 6(-7k - 9) \quad \text{(by distributing the -6 to get -42k - 54)} \]

In this case, the response options provided don't exactly match our factorization.

However, if we had to select an option based on the closest representation from the responses you provided, we would select:

\(-6(7k + 9)\).

If there are further clarifications needed, feel free to ask!

Factor the expression completely.

100x+1000


(1 point)
Responses

10(10x+100)
10 times open paren 10 x plus 100 close paren

100(x+10)
100 times open paren x plus 10 close paren

100(x+100)
100 times open paren x plus 100 close paren

1000(10x+1)

To factor the expression \(100x + 1000\) completely, we start by finding the greatest common factor (GCF) of the terms.

  1. The coefficients are 100 and 1000. The GCF is 100.
  2. We can factor out 100 from the expression:

\[ 100x + 1000 = 100(x + 10) \]

So the correct factored form of the expression \(100x + 1000\) is:

\[ \boxed{100(x + 10)} \]

Based on the provided responses, the correct answer is:

100(x + 10), which corresponds to the second response option.

Simplify.

3−(4x−5)+6


(1 point)
Responses

−4x+14
negative 4 x plus 14

4x+4
4 x plus 4

12x+21
12 x plus 21

−4x+4

To simplify the expression \(3 - (4x - 5) + 6\), follow these steps:

  1. Distribute the negative sign in front of the parentheses:

    \[ 3 - (4x - 5) = 3 - 4x + 5 \]

  2. Combine like terms (the constants in this case):

    \[ 3 + 5 - 4x = 8 - 4x \]

  3. Now, add the \(6\) that remains from the original expression:

    \[ 8 - 4x + 6 = 14 - 4x \]

  4. Finally, rearranging it gives us:

    \[ -4x + 14 \]

So the simplified expression is:

\[ \boxed{-4x + 14} \]

This matches the first response option.

Ella Grace bought 5 plants for her backyard for $20 each and 3 chairs for her firepit for $20 each. She used this expression to calculate the total amount she spent.

(5⋅20)+(3⋅20)


What is another expression to calculate the total amount spent?

(1 point)
Responses

5⋅(20+3)
5 times open paren 20 plus 3 close paren

20⋅(5+3)
20 times open paren 5 plus 3 close paren

(5+20)⋅(3+20)
open paren 5 plus 20 close paren times open paren 3 plus 20 close paren

5⋅20⋅3

To find another expression that calculates the total amount spent by Ella Grace for her plants and chairs, we can consider the expression she originally used:

\[ (5 \cdot 20) + (3 \cdot 20) \]

We can factor out the common term \(20\):

\[ (5 + 3) \cdot 20 \]

This means that she bought a total of \(5 + 3 = 8\) items, each costing \(20\). So we can rewrite the expression as:

\[ 20 \cdot (5 + 3) \]

Thus, the equivalent expression to calculate the total amount spent is:

\[ \boxed{20 \cdot (5 + 3)} \]

This corresponds to the second response option.