Factorization - Common Factor: Matching expressions equal in value

Let's solve the problem step by step:

• Step 1: Identify the greatest common factor (GCF) of the coefficients in the expression $8x^2 + 4x$. The numbers are 8 and 4, and the GCF is 4.

• Step 2: Factor out the common variable. Both terms have $x$ as a common variable factor, so the GCF of the variable part is $x$.

• Step 3: Factor the expression using the GCF. We take $4x$ as a common factor from both terms:
      $8x^2$ can be rewritten as $4x \cdot 2x$.
      $+4x$ can be rewritten as $4x \cdot 1$.

• Step 4: Write the factored expression:
      $8x^2 + 4x = 4x(2x + 1)$.

• Step 5: Verify by checking each option. The expression we obtained $4x(2x + 1)$ matches the choice with 2.

Therefore, the equivalent expression is $4x(2x+1)$.

To solve the problem, the goal is to factor the expression $15z^2 + 50zx$ by finding the greatest common factor.

Step 1: Identify the greatest common factor (GCF):

• The terms in the expression are $15z^2$ and $50zx$.
• The numerical coefficients are $15$ and $50$, and their GCF is $5$.
• Both terms contain $z$ as a factor, so $z$ is also part of the GCF.
• Therefore, the GCF of both terms is $5z$.

Step 2: Factor the expression using the GCF:

• Divide each term by the GCF $5z$:
• $\frac{15z^2}{5z} = 3z$
• $\frac{50zx}{5z} = 10x$
• Thus, the expression can be rewritten as the product of the GCF and the remaining terms: $5z(3z + 10x)$.

Therefore, the expression $15z^2 + 50zx$ is equivalent to $5z(3z + 10x)$.

To solve this problem, we'll focus on factorization by grouping:

• Step 1: Identify groupable terms in $7z + 10b + 2bz + 35$.
• Step 2: Reorganize and group terms for greatest common factor extraction.
• Step 3: Factor each group and simplify.

Now, let's work through each step:

Step 1:
Observe that we can reorganize the expression to facilitate grouping:

$7z + 35 + 10b + 2bz$.

Step 2:
Group into pairs: $(7z + 35) + (10b + 2bz)$.
Within each pair, extract common factors:

$= 7(z + 5) + 2b(z + 5)$, noticing that each group factors nicely.

Step 3:
Since both terms now have a common factor of $(z + 5)$, we can factor it out:

$= (7 + 2b)(z + 5)$.

Therefore, the expression $7z + 10b + 2bz + 35$ is equivalent to $(7 + 2b)(z + 5)$.

This matches choice 1: $(7+2b)(z+5)$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the greatest common factor (GCF) in the expression.

• Step 2: Factor out the GCF from the expression.

• Step 3: Compare the factored expression with the choices provided.

Now, let's work through each step:
Step 1: The expression given is $2xy + x^2 + 3x$. The GCF of these terms is $x$ because it appears in each term.
Step 2: Factor out $x$ from each term, which gives:$x(2y + x + 3)$ This rewrites the expression in its factored form.
Step 3: Compare the factored form $x(2y + x + 3)$ with the answer choices.

Choice 1: $2x + 2y + 3$ does not match the factored form.
Choice 2: $x(2y + x + 3)$ exactly matches the factored form.
Choice 3: $2x(y + 4)$ does not match the factored form.
Choice 4: $2x(y + x + 3)$ does not match the factored form.

Therefore, the expression $2xy + x^2 + 3x$ is equivalent to $x(2y + x + 3)$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the greatest common factor (GCF) of the coefficients in the expression.

• Step 2: Factor out the GCF from the entire expression.

• Step 3: Confirm that the factored expression matches one of the given answer choices.

Now, let's work through each step:
Step 1: The coefficients of the terms $99ab^2$ and $81b$ are 99 and 81, respectively. The greatest common factor of these coefficients is 9.
Step 2: Both terms, $99ab^2$ and $81b$, contain the variable $b$. We can factor out $b$ as well. Thus, the GCF of both terms in the expression $99ab^2 + 81b$ is $9b$.
Step 3: Factor $9b$ out of both terms:
$\begin{aligned} 99ab^2 + 81b &= 9b( \frac{99ab^2}{9b} + \frac{81b}{9b} )\\ &= 9b(11ab + 9) \end{aligned}$
Therefore, the equivalent expression to the given algebraic expression is $9b(11ab + 9)$. This matches choice 2.

Thus, the solution to the problem is $9b(11ab + 9)$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the common factor in the expression.
• Step 2: Factor out the common factor using the distributive property.
• Step 3: Simplify the expression inside the parentheses.

Now, let's work through each step:

Step 1: Identify the common factor. The given expression is $2a(b+3) + 4(b+3)$. Notice that both terms have a common factor, which is $(b+3)$.

Step 2: Factor out the common factor. Using the distributive property in reverse, we can factor out $(b+3)$:

Step 3: Simplify the expression inside the parentheses if needed. In this case, $2a + 4$ is already simplified.

Therefore, the expression $2a(b+3) + 4(b+3)$ simplifies to the equivalent expression $(b+3)(2a+4)$.

The correct choice that corresponds to this expression is choice 3: $(b+3)(2a+4)$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the greatest common factor (GCF) of the terms in the expression $16 - 4c$.
• Step 2: Factor out the GCF from both terms.
• Step 3: Compare the factored expression with the provided choices to find the equivalent expression.

Let's work through each step:

Step 1: Identify the GCF.
The terms in the expression are $16$ and $-4c$. The greatest common factor of $16$ and $4$ (coefficient of $c$) is $4$.

Step 2: Factor out the GCF.
Factor $4$ out from each term in the expression:
$16 - 4c = 4 \times 4 - 4 \times c = 4(4 - c)$.

Step 3: Compare with the choices.
We factorized $16 - 4c$ to get $4(4 - c)$. Now we compare it with the provided choices:

• Choice 1: $4(12 - c)$ does not match our expression.
• Choice 2: $4(4 - 4c)$ is also not equivalent.
• Choice 3: $4(2 - c)$ is not equivalent.
• Choice 4: $4(4 - c)$ matches our expression precisely.

Therefore, the expression equivalent to $16 - 4c$ is $\boxed{4(4 - c)}$.

To solve the problem, we'll follow these steps:

• Step 1: Identify the common factor in the expression $6mn+\frac{3n}{m}+9n^2$.
• Step 2: Factor this common factor out from the expression.
• Step 3: Compare the factored expression to the given choices.

Let's work through each step:

Step 1: Identify the Common Factor

Looking at the terms $6mn$, $\frac{3n}{m}$, and $9n^2$, the common factor among them is clearly $3n$ since:

• $6mn$ can be divided by $3n$, giving $2m$.
• $\frac{3n}{m}$ can be divided by $3n$, giving $\frac{1}{m}$.
• $9n^2$ can be divided by $3n$, giving $3n$.

Step 2: Factor Out the Common Factor

Factoring $3n$ out of each term, we rewrite the expression:

$6mn + \frac{3n}{m} + 9n^2 = 3n(2m) + 3n\left(\frac{1}{m}\right) + 3n(3n)$.

This simplifies to:

$3n(2m + \frac{1}{m} + 3n)$.

Step 3: Compare with Choices

We compare our factored expression, $3n(2m + \frac{1}{m} + 3n)$, to the given choices. We find that Choice 1 matches our factored form.

Therefore, the expression $6mn+\frac{3n}{m}+9n^2$ is equivalent to $3n(2m+\frac{1}{m}+3n)$.

Let's determine the equivalence of different expressions to $2ab-4bc$ by factorization:

Step 1: Factor the given expression:

The expression $2ab - 4bc$ has a common factor of $2b$.

Factor out $2b$, we obtain:

$\begin{aligned} 2ab - 4bc &= 2b(a) - 2b(2c) \\ &= 2b(a - 2c) \end{aligned}$

Step 2: Compare with each option:

• Option 1: $2b(a-2c)$

  • This is identical to the factorized form $2b(a-2c)$, so it is equivalent.

• Option 2: $2b(-2c+a)$

  • Although it appears reversed, $-2c + a$ is equivalent to $a - 2c$, so it's equivalent.

• Option 3: $2(-2bc+ab)$

  • By rearranging:

    $\begin{aligned} 2(-2bc + ab) &= 2(ab - 2bc) \\ &= 2ab - 4bc \end{aligned}$
    

  • It matches the original expression, thus is equivalent.

• Option 4: $2a(2bc-b)$

  • Expanding:

    $2a(2bc-b) = 4abc - 2ab$
    

  • Does not match $2ab - 4bc$, so not equivalent.

Conclusion: The expressions equivalent to $2ab - 4bc$ are Options 1, 2, and 3.

Therefore, the solution to the problem is $1,2,3$.

Let's start by factoring the given expression $12n + 48 - 36mn - 144m$.

First, notice that:

• In the first two terms: $12n + 48$, we can factor out $12$, giving us $12(n + 4)$.

• In the last two terms: $-36mn - 144m$, we can factor out $-36m$, resulting in $-36m(n + 4)$.

Combining both factorizations, we can write the original expression as:

$12(n + 4) - 36m(n + 4)$

Now, we factor out the common term $(n + 4)$:

$(12 - 36m)(n + 4)$

Thus, the expression simplifies to:

$12(1 - 3m)(n + 4)$

Now, let's verify which options match:

Option 1: $12(1-3m)(n+4)$
This directly matches our simplified expression, so it is a correct choice.

Option 2: $3(4 - 12m)(n + 4)$
Simplifying: Factoring 3 from $4 - 12m$ gives $3 \times 4(1 - 3m)$, which matches the expression $12(1 - 3m)(n + 4)$. So, this is also a correct choice.

Option 3: $12(-3m + 4)(n + 1)$
The factors do not align with our expression because $(n+1)$ is not factored from $(n + 4)$.

Option 4: $12(n + 4) - 3m(12n + 48)$
Rewriting: $12(n+4) - 3m(12(n+4)) = (12 - 36m)(n+4)$ which matches. Therefore, it is correct.

Hence, options 1, 2, and 4 are equivalent to the original expression.

The correct answer to the problem is

$1, 2, 4$

To solve this problem, let's simplify and factor the given expression:

Original expression: $\frac{36t}{r} - 18rt^2$

Step 1: Factor out the greatest common divisor, which is 18:

$18 \left( \frac{2t}{r} - rt^2 \right)$

This matches the structure of choice 1.

Step 2: Substitute this back into the given options and simplify.

Option 1: $18\left( \frac{2t}{r} - rt^2 \right)$

This is identical to the factored form of the original.

Option 2: $18r\left( \frac{2t}{r^2} - t^2 \right)$

Expand and simplify:

$= 18r \cdot \frac{2t}{r^2} - 18r \cdot t^2$

$= \frac{36t}{r} - 18rt^2$

This matches the original expression.

Option 3: $-18(rt + \frac{2t}{r})$

Expand and simplify:

$= -18rt - \frac{36t}{r}$

This does not match the original expression.

Option 4: $6t\left( \frac{6t}{r} - 3rt \right)$

Expand and simplify:

$= 6t \cdot \frac{6t}{r} - 6t \cdot 3rt$

$= \frac{36t^2}{r} - 18rt^2$

This does not match the original expression.

Therefore, the correct options that are equivalent to the given expression are 1 and 2.

To solve this problem, we'll simplify and compare the given expressions. Let's begin by factoring the original expression:

Factor the original expression $\frac{14m}{n} + 21m^2$:

Notice both terms contain a factor of $7m$:

$\frac{14m}{n} + 21m^2 = 7m\left(\frac{2}{n} + 3m\right)$

Now, let's examine each given choice:

• Choice 1: $7m\left(\frac{2}{n} + 3m\right)$ matches the factored version we derived from the original, so they are equivalent.
• Choice 2: Simplifying $7\left(\frac{2}{nm} + 3m^2\right)$:

  $= 7 \cdot \frac{2}{nm} + 7 \cdot 3m^2 = \frac{14}{nm} + 21m^2$; note this is different from the original expression since we require $\frac{14m}{n}$ in the first term.

• Choice 3: Simplicity is tested for $\frac{m}{n}(14 + 3m)$:

  $= \frac{14m}{n} + \frac{3m^2}{n}$; for equivalence, recall we needed complete $\frac{14m}{n} + 21m^2$, not $\frac{3m^2}{n}$

• Choice 4: Examine $7\frac{m}{n}(2 + 3mn)$:

  Bifurcation gives $= \frac{14m}{n} + 21m^2$; multipliers yield the original expression accurately.

Thus, the answer is clearly seen that both choices 1 and 4 are equivalent to the original expression:

The correct choices are 1 and 4.

To solve this problem, we need to systematically expand and simplify each expression given in the problem statement:

• Expression 1: $(6b+3)(-2+a)$

  Expand using the distributive property:

  $= 6b(-2) + 6b(a) + 3(-2) + 3(a)$

  $= -12b + 6ab - 6 + 3a$

  Reorder terms: $6ab + 3a - 12b - 6$

• Expression 2: $(2b+1)(3a-6)$

  Expand using the distributive property:

  $= 2b(3a) + 2b(-6) + 1(3a) + 1(-6)$

  $= 6ab - 12b + 3a - 6$

  This simplifies directly to $6ab + 3a - 12b - 6$

• Expression 3: $(a+3)(6b-2)$

  Expand using the distributive property:

  $= a(6b) + a(-2) + 3(6b) + 3(-2)$

  $= 6ab - 2a + 18b - 6$

  This results in $6ab - 2a + 18b - 6$, clearly different from the others

• Expression 4: $6ab+3a-12b-6$

  This is already simplified and the same as the results of expressions 1 and 2.

Upon comparing the simplified expressions, expressions 1, 2, and 4 have the same value: $6ab + 3a - 12b - 6$. Expression 3 differs with $6ab - 2a + 18b - 6$.

Thus, the expressions with the same value are 1, 2, and 4.

Therefore, the correct answer is choice 4: $1,2,4$.