Factorization - Common Factor: Applying the formula

To solve this problem, we'll employ the method of factoring by common factor:

• Step 1: Identify the common factor in the terms $2ax$ and $4x^2$.
• Step 2: Factor out the common factor.
• Step 3: Verify the factored form by expanding to ensure it equals the original expression.

Let's begin with Step 1:

Looking at the terms $2ax$ and $4x^2$, we can see that both terms include $x$ as a common factor. They also share the coefficient '2'. Therefore, the greatest common factor (GCF) is $2x$.

In Step 2, we'll factor $2x$ out of each of the terms:

$2ax$ becomes $2x \cdot a$, and
$4x^2$ becomes $2x \cdot 2x$.

So the expression can be written as:

$2ax + 4x^2 = 2x(a) + 2x(2x)$

Thus, factored completely using the common factor $2x$:

$2ax + 4x^2 = 2x(a + 2x)$

In Step 3, let’s verify by expanding:

Expanding $2x(a + 2x)$, we have:

$2x \cdot a + 2x \cdot 2x = 2ax + 4x^2$,
which confirms our factorization is correct.

Therefore, the solution to the problem is $2x(a + 2x)$.

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

• Step 1: Identify any common factors in the terms provided.
• Step 2: Factor the expression by extracting the common factor.

Now, let's work through each step:
Step 1: The expression given is $2ax + 3x$. We observe that both terms have a common factor of $x$.
Step 2: We factor out $x$ from each term, which results in $x(2a + 3)$, because:

• $2ax$ divided by $x$ results in $2a$.
• $3x$ divided by $x$ results in $3$.

Therefore, the expression $2ax + 3x$ can be factored as $x(2a+3)$.

$ab+bc=a\times b+b\times c$

If we consider that b is the common factor, it can be removed from the equation:

$b(ab+bc)=$

We divide by b:$b(\frac{ab}{b}+\frac{bc}{b})=$

$b(a+c)$

We divide 14 into a multiplication exercise to help us simplify the calculation accordingly:$7\times a+7\times b\times2=$

We then extract the common factor 7:

$7(a+2\times b)=7(a+2b)$

We begin by breaking down the coefficients 12 and 16 into multiplication exercises with a multiplying factor to eventually simplify:

$3\times4\times x+4\times4\times y$

We then extract 4 which is the common factor:

$4(3\times x+4\times y)=4(3x+4y)$

To find the common factor for the expression $22abc - \frac{11ab}{c}$, we will follow these steps:

• Step 1: Determine the common factors in terms of coefficients and variables.
• Step 2: Factor these common elements out of the expression.
• Step 3: Simplify the expression after factoring.

Now, let's proceed with the solution:

Step 1: Identify the common factors.
The expression is $22abc - \frac{11ab}{c}$. We can see that both terms have the factors $11ab$. 11 is a common numerical factor in 22 and 11. Ab is also present in both terms.

Step 2: Factor out the common factors.
Factor out $11ab$ from each term in the expression:

Step 3: Simplify the terms inside the parenthesis.

• Simplify the first term: $\frac{22abc}{11ab} = 2c$
• Simplify the second term: $\frac{\frac{11ab}{c}}{11ab} = \frac{1}{c}$

Thus, we get the factored form:

Therefore, the common factor form of $22abc - \frac{11ab}{c}$ is $11ab(2c - \frac{1}{c})$.

First, we will decompose the coefficients of the multiplication exercise that will help us find the common factor:

$25\times y-4\times25\times x\times y\times y$

Now find the common factor 25y:

$25y(1-4xy)$

To solve this problem, we'll follow these steps to factor $256xy^3 - 32zy^2$:

• Step 1: Identify the greatest common factor (GCF) for both terms.
• Step 2: Factor out the GCF from the expression.

Now, let's work through each step:

Step 1: Identify the GCF.
The first term is $256xy^3$, and the second term is $32zy^2$.

• The numerical GCF of 256 and 32 is 32.
• The terms $xy^3$ and $zy^2$ have a common factor of $y^2$.

Thus, the GCF of the entire expression is $32y^2$.

Step 2: Factor out the GCF.
We'll factor $32y^2$ out of each term:

• The first term $256xy^3$ becomes $32y^2 \times 8xy$. (since $\frac{256xy^3}{32y^2} = 8xy$)
• The second term $32zy^2$ becomes $32y^2 \times z$. (since $\frac{32zy^2}{32y^2} = z$)

Therefore, factoring the expression by the GCF gives us:
$256xy^3 - 32zy^2 = 32y^2(8xy - z)$.

The factorized form of the expression is $32y^2(8xy - z)$.

To factor the expression $26a + 65bc$, we start by identifying the greatest common factor of the coefficients:

• The coefficients are 26 and 65.
• The prime factorization of 26 is $2 \times 13$.
• The prime factorization of 65 is $5 \times 13$.
• The greatest common factor of 26 and 65 is 13, because both coefficients are divisible by 13.

Now, factor out $13$ from both terms of the expression:

$26a = 13 \times 2a$

$65bc = 13 \times 5bc$

So, we can express the entire expression as:

$26a + 65bc = 13(2a + 5bc)$

Therefore, the factorized form of the expression is $\mathbf{13(2a + 5bc)}$.

Comparing with the given choices, this corresponds to choice 1.

Let's factor the given expression:

$37a+6b$We will do this by extracting the greatest common factor, both for numbers and letters,

We will address numbers and letters separately, remembering that a common factor is a factor (multiplier) that is common to all terms in the expression,

Let's start with the numbers:

Note that for the two numerical coefficients of the terms in the expression, namely the numbers 6 and 37, there is no single factor that is common to both, while the factors of the number 6 are the numbers 2 and 3 or 6 and 1, the number 37 is a prime number and therefore its only factors are 37 and 1, meaning - there is no factor that is common to these two numbers, therefore - the number 1 (which is essentially the 0 power of any number - except 0) will be considered instead of the common factor for numbers.

For the letters:

There are two terms:
$a,\hspace{4pt}b$It's easy to see that there is no factor common to these two terms,$a$therefore there is no algebraic expression for letters that could be a common factor, meaning - the number 1 (which is essentially the 0 power of any number - except 0) will be considered instead of the common factor for letters.

Therefore we conclude:

This expression cannot be factored by extracting a common factor (or in any other way)

Therefore the correct answer is answer D.

We first break down the coefficient of 20 into a multiplication exercise. That will help us to simplify the calculation :$5\times4\times a\times b-4\times a\times c$

We then extract 4a as a common factor:$4a(5\times b-c)=4a(5b-c)$

To solve the problem of factorizing $36mn - 60m$, we will follow these steps:

• Step 1: Identify the Greatest Common Factor (GCF) of the coefficients and the variable part.

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

• Step 3: Simplify and verify the factorized expression.

Step 1: Identify the GCF

We begin by finding the GCF of the numerical coefficients 36 and 60. The prime factorizations are:

• $36 = 2^2 \times 3^2$

• $60 = 2^2 \times 3 \times 5$

The GCF for 36 and 60 is $2^2 \times 3 = 12$.

Next, consider the variable part $m$, which appears in both terms of the expression. Thus, the GCF of the entire expression is $12m$.

Step 2: Factor out the GCF

We factor $12m$ out of each term:

$\begin{aligned} 36mn - 60m &= 12m(3n) - 12m(5) \\ &= 12m(3n - 5) \end{aligned}$

Step 3: Simplify and Verify

The factored expression is $12m(3n - 5)$. Expanding this back verifies the factorization:

$\begin{aligned} 12m(3n - 5) &= 12m \cdot 3n + 12m \cdot (-5) \\ &= 36mn - 60m \end{aligned}$

which matches the original expression, confirming our factorization is correct.

Therefore, the factorized form of the given expression is $12m(3n - 5)$.

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

• Step 1: Identify the greatest common factor (GCF) of the terms in the expression.
• Step 2: Factor out the GCF from the expression.
• Step 3: Simplify the expression inside the parentheses.

Step 1: Determine the GCF of the terms $13abcd$ and $26ab$.
The GCF of the coefficients $13$ and $26$ is $13$.
The variables $a$ and $b$ are common in both terms, so the GCF of the variables is $ab$.

Therefore, the GCF of the expression is $13ab$.

Step 2: Factor out the GCF.
Factor $13ab$ from each term in the expression:
$13abcd + 26ab = 13ab(cd) + 13ab(2)$

Step 3: Simplify to obtain the final factorised expression:
$13ab(cd + 2)$

Therefore, the factorised form of the expression is $13ab(cd + 2)$.

Among the given choices, this corresponds to choice 2: $13ab(cd+2)$.

First, we break down all the powers into multiplication exercises and at the same time try to reduce the integers as much as possible:

7*2*xyz+2*4*x*x*y*y²*z

Now we use the substitution property to arrange the equation into a more manageable form:

2*x*y*z*7+2*x*y*z*x*y²

Lastly we try to find the common factor among all the parts - 2xyz

2xyz(7+xy²)

Breakdown into factors by groups The given expression:

$(x+8)(2y+16)+4(3x+24)$This is done by extracting a common factor, both for the numerators and for the letters, in one of the parts of the given expression (the parts of the expression are separated by addition or subtraction operations between the multiplication terms) or both, in order to be able to distinguish a common multi-variable factor, and take it out of the parentheses, etc.

Let's refer separately to the numerators and letters, recalling that a common factor is a factor (multiplier) common to all the terms in the expression,

Let's start by examining the two parts of the expression separately, the first expression on the left:

$(x+8)(2y+16)$and the second expression on the left:

$4(3x+24)$Note that the expression in the parentheses in the second expression above and the expression in the parentheses in the first expression on the left in the original expression are proportional to each other (i.e. it is possible to get from one expression to the other by multiplication by some factor), this is because it is possible to take out a common factor out of the parentheses:

$4(3x+24) \\ 4\cdot3(x+8) \\ 12(x+8)$

$$We used the fact that the number 24 is a multiple of the number 3:

$24=8\cdot3$

Let's now go back to the original expression in the question and apply this knowledge:

$(x+8)(2y+16)+4(3x+24) \\ \downarrow\\ (x+8)(2y+16)+4\cdot3(x+8)\\ (x+8)(2y+16)+12(x+8)$Let's now use the distributive property and rearrange the expression we got again:

$(x+8)(2y+16)+12(x+8) \\ (2y+16)(x+8)+12(x+8)$Now we can note that in the expression we got in the last step there is a common factor which is multi-variable (i.e. includes more than one variable in the expression):

$x+8$This is because it is a multiple of both the first part of the expression on the left (the parentheses multiplier) and of the second part of the expression on the left:

$(2y+16)\underline{(x+8)}+12\underline{(x+8)}$Therefore, we can take out this expression in its entirety out of the parentheses as a common factor and break down the given expression in the usual way (i.e. using the answer to the question: "By what can we multiply the common factor (including its sign) in order to get each of the terms in the original expression (including their sign)?"):

$\textcolor{red}{ (2y+16)(x+8)}\textcolor{blue}{+12(x+8) } \\ \underline{(x+8)}\cdot\textcolor{red}{(2y+16)}+\underline{(x+8)}\cdot\textcolor{blue}{(+12)}\\ \downarrow\\ \underline{(x+8)}\big(\textcolor{red}{(2y+16)}\textcolor{blue}{+12}\big)$

In the expression above the operation is explained using colors and signs:

The common factor is highlighted using an underline, and the multipliers inside the parentheses correspond to the terms in the original expression using colors, note that we also referred to the sign, both of the common factor (in black) that we took out of the parentheses and to the signs of the terms in the original expression (in colors), there is no need to display this in steps as described above, you can (and should) jump directly to the broken down form in the last line, but definitely need to refer to the signs above, as in each term the sign is an integral part of it,

So we got the expression broken down into factors (by groups):

$(x+8)\big((2y+16)+12\big)$Let's continue and complete the breakdown while simplifying the expression in the right parentheses, note that the expression we got in the right parentheses in the multiplier of the parentheses we got in the last step, can be further broken down into factors by taking out the common factor the number: 2, this is because the number 28 is a multiple of the number 2:

$(x+8)(2y+28) \\ (x+8)\cdot2(y+14) \\ 2(x+8)(y+14)$In the last step we used the distributive property to rearrange the expression we got.

Let's summarize the steps of breaking down the expression (by groups), we got that:

$(x+8)(2y+16)+4(3x+24) \\ (x+8)(2y+16)+12(x+8) \\ (x+8)\big((2y+16)+12\big) \\ (x+8)(2y+28) \\ 2(x+8)(y+14)$We can be sure that this breakdown is correct easily by opening the parentheses using the extended distribution law and verifying that indeed the original expression we broke down is obtained term by term, this should be done while paying attention to the signs of the terms in the original expression and to the sign (given for selection always) of the common factor.

Therefore, the correct answer is answer c.

Note that in the given expression there are two completely different terms, meaning - the letters cannot be factored out, so we will factor out the greatest common factor of the numbers 6 and 3, which is clearly the number 3 and is a factor of both other numbers:$3x^3+6a^4 =3(x^3+2a^4)$After factoring out the common factor outside the parentheses, we will look at each term before factoring out the common factor separately, asking ourselves: "By how much did we multiply the common factor to get the current term?" and fill in the missing parts inside the parentheses while making sure that the sign of the term we completed inside the parentheses when multiplied by the sign of the term we factored out will give us the sign of the original term, it is recommended to verify that the factoring was done correctly by opening the parentheses, performing the multiplications and confirming that we indeed get the expression before factoring.

Therefore, the correct answer is answer B.

To solve this problem, we will follow the steps below:

Let's start by examining the given expression:

$\frac{2x}{b} + \frac{4c}{3b}$

Step 1: Identify the common factor in both terms.

Both terms have a denominator of $b$. We can factor $\frac{1}{b}$ out of the entire expression.

Step 2: Factor out the common factor.

$\frac{2x}{b} = \frac{1}{b} \cdot 2x$
$\frac{4c}{3b} = \frac{1}{b} \cdot \frac{4c}{3}$

Step 3: Combine terms using the common factor.

Now, factor $\frac{1}{b}$ out of the expression:
$\frac{1}{b}(2x + \frac{4c}{3})$

Step 4: Simplify the expression.

We notice that both terms have a common factor of 2 in the numerators:

$\frac{1}{b} \cdot (2x + \frac{4c}{3}) = \frac{2}{b} \cdot (x + \frac{2c}{3})$

Therefore, the common factor in the expression is $\frac{2}{b}(x+\frac{2c}{3})$.

To solve this problem, we will follow these steps:

• Step 1: Determine the greatest common divisor (GCD) of the numeric coefficients 18 and 99.
  The GCD of 18 and 99 is 9 since $18 = 2 \times 9$ and $99 = 11 \times 9$.
• Step 2: Analyze the variables in the terms $18\frac{ab}{c^2}$ and $-99\frac{c^2}{ab}$.
  Both terms have the fraction form with $\frac{ab}{c^2}$ and $-\frac{c^2}{ab}$.
• Step 3: Identify the common algebraic fraction factor.
  The common factor from the algebraic component is $\frac{ab}{c^2}$.
• Step 4: Factor $\frac{ab}{c^2}$ out from the expression.
  The expression $18\frac{ab}{c^2} - 99\frac{c^2}{ab}$ can be rewritten as:
  $9\frac{ab}{c^2}(2 - 11\frac{c^4}{a^2b^2})$.

Therefore, the decomposition of the given expression is $9\frac{ab}{c^2}(2 - 11\frac{c^4}{a^2b^2})$.

To solve the problem, we aim to factor the expression by identifying the greatest common factor (GCF) for the terms $15xyz^2$ and $25\frac{xy}{z}$.

Step 1: Identify the GCF
- Both terms contain the factors $x$ and $y$.
- The first term $15xyz^2$ consists of $15$, $x$, $y$, and $z^2$.
- The second term $25\frac{xy}{z}$ consists of $25$, $x$, $y$, and $\frac{1}{z}$.
- The GCF of the constants 15 and 25 is 5.

Step 2: Factor out the GCF
- The GCF of the variables is $xy$.
- Therefore, the overall GCF we can factor out is $5xy$.

Step 3: Simplify the remaining expression
- Factoring out $5xy$ from the expression:
$15xyz^2 = 5xy \times 3z^2$
$25\frac{xy}{z} = 5xy \times \frac{5}{z}$

Step 4: Write the factored expression
This gives us:
$15xyz^2 + 25\frac{xy}{z} = 5xy(3z^2 + \frac{5}{z})$

Thus, the decomposition of the simplified expression is $5xy(3z^2 + \frac{5}{z})$, which corresponds to choice 2.

Therefore, the solution to the problem is $5xy(3z^2 + \frac{5}{z})$.

To factor the expression $\frac{xy}{2}+\frac{x}{4y}$, we proceed as follows:

• Step 1: Identify the common factor between the terms.
• Step 2: Factor out the common factor.
• Step 3: Simplify the expression inside the parentheses.

Let's break this down:
Step 1: The expression is $\frac{xy}{2} + \frac{x}{4y}$. Clearly, both terms share $\frac{x}{2}$ as a common factor.
Step 2: Factor out $\frac{x}{2}$ from each term:
- From the first term: $\frac{xy}{2} = \frac{x}{2} \times y$.
- From the second term: $\frac{x}{4y} = \frac{x}{2} \times \frac{1}{2y}$.
Step 3: This gives us:
$\frac{xy}{2} + \frac{x}{4y} = \frac{x}{2}(y + \frac{1}{2y})$

Thus, the expression can be decomposed into factors as $\frac{x}{2}(y+\frac{1}{2y})$.