Factorization - Common Factor: Number of terms

Factor the given expression:

$15a^2+10a+5$We will do this by factoring out the greatest common factor, both from the numbers and the letters,

We must refer to the numbers and letters separately, remembering that a common factor is a factor (multiplier) common to all the terms of the expression,

Let's start with the numbers

Note that the numerical coefficients of the terms in the given expression, that is, the numbers: 5,10,15 are all multiples of the number 5:

$15=3\cdot\underline{5}\\ 10=2\cdot\underline{5}\\$Therefore, the number 5 is the greatest common factor of the numbers,

For the letters:

Note that only the first two terms on the left depend on x, the third term is a free number that does not depend on x, hence there is no common factor for all three terms together for the letters (that is, we will consider the number 1 as the common factor for the letters)

Therefore, we can summarise as follows:

The greatest common factor (for numbers and letters together) is:

$5\cdot1\\ \downarrow\\ 5$Let's take the above value as a multiple outside the parenthesis and ask the question: "How many times must we multiply the common factor (including its sign) in order to obtain the terms of the original expression (including their signs)?" Using this method we can determine what is the expression inside the parenthesis that multiplied the common factor:

$\textcolor{red}{ 15a^2}\textcolor{blue}{+10a} \textcolor{green}{+5} \\ \underline{5}\cdot\textcolor{red}{3a^2}+\underline{5}\cdot\textcolor{blue}{(+2a)}+\underline{5}\cdot\textcolor{green}{(+1)}\\ \downarrow\\ \underline{5}(\textcolor{red}{3a^2}\textcolor{blue}{+2a}\textcolor{green}{+1})$In the previous expression, the operation is explained through colors and signs:

The common factor has been highlighted with an underscore, and the multiples inside the parenthesis are associated with the terms of the original expression with the help of colors.

Note that in the detail of the decomposition above, we refer both to the sign of the common factor (in black) that we extracted as a multiple outside of the parenthesis, as well as to the sign of the terms in the original expression (in colors) Note that there is no obligation to show it. Whilst the above method is described in stages it is recommended to jump directly to the broken down form in the last line whilst continuing to refer to the previous signs which are an integral part of the expression.

We can easily ensure that this decomposition is correct by opening the parentheses with the help of the distributive property. As such we can ensure that the original expression that we decomposed can be effectively recovered - Remember, this must be done emphasizing the sign of the members in the original expression as well as the sign (which is always selectable) of the common factor.

(Initially, you should use the previous colors to ensure you get all the terms and multiples in the original expression; later on, it is recommended not to use the colors)

Therefore, the correct answer is option b.

Factor the given expression:

$xyz+yzt+ztw+wtr$
We will do this by extracting the highest common factor, both from the numbers and the letters.

We refer to the numbers and letters separately, remembering that a common factor is a factor (multiplier) common to all terms of the expression.

As the given expression does not have numeric coefficients (other than 1), we will look for the highest common factor of the letters:

There are four terms in the expression:
$xyz,\hspace{4pt}yzt,\hspace{4pt}ztw,\hspace{4pt}wtr$We will notice that in each of the four members there are three different letters, but there is not one or more letters that are included (in the multiplication) in all the terms; that is, there is no common factor for the four terms and therefore it is not possible to factor this expression by extracting a common factor.

Therefore, the correct answer is option d.

Factor the given expression:

$4a+13b+58c$We will do this by extracting the greatest common factor, both from the numbers and the letters,

We will refer to the numbers and letters separately, remembering that a common factor is a factor (multiple) common to all terms of the expression,

Let's start with the numbers:

We first notice that the numerical coefficients of the terms in the given expression, that is, the numbers 4, 13, 58, do not have a common factor, and this is because the number 13 is a prime number and the other two numbers are not multiples of it,

Therefore, there is no common factor for the numbers hence we select the number 1, as the common factor for the numbers (Reminder: a number raised to the power zero is always given to be equal to one)

For the letters:

There are three terms in the expression:
$a,\hspace{4pt}b,\hspace{4pt}c$It is easy to see that there is no common factor for these three terms,

Hence, it is not possible to factor the given expression with the help of a common factor.

Therefore, the correct answer is option d.

We will factorize by extracting the largest common factor. In the given expression, there are the following terms:

$$$2a^5,\hspace{4pt}8a^6,\hspace{4pt}4a^3$

We'll start with the letters, using the law of exponents for multiplication between terms with identical bases, but in reverse order:

$b^{m+n} =b^m\cdot b^n$

to understand that:

$a^6=a^3\cdot a^3\\ a^5=a^3\cdot a^2$

and we'll note that the highest power of $a$that can be extracted as a common factor for all three terms is the power of 3, meaning: $a^3$,

we'll continue and extract the largest common factor for the numbers 2,4,8, which is clearly the number 2 since it is a prime number, therefore we'll conclude and extract the common factor: $2a^3$

We'll apply this to the given expression and extract a common factor:

$2a^5+8a^6+4a^3 =2a^3(a^2+4a^4+2)$

where after extracting the common factor outside the parentheses, we'll look at each term before extracting the common factor separately, asking the question: "By how much did we multiply the common factor to get the current term?" and we'll 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 extracted outside the parentheses gives us the sign of the original term, it is recommended to verify that the factorization is correct by opening the parentheses, performing the multiplications and confirming that we indeed get the expression before factorization.

Therefore the correct answer is answer A.

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

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

Now, let's work through each step:
Step 1: The numerators of the terms are $ab$, $a^2b$, and $ab^2$. The GCF here is $ab$.

Step 2: Factor $ab$ from the expression:

Step 3: Factor $\frac{1}{cd}$ from the expression inside the parenthesis to simplify further:

Therefore, the solution to the problem is $\frac{ab}{cd}(\frac{1}{d}+\frac{a}{c}+\frac{b}{d^2})$.

To factor the given expression, we proceed as follows:

• Step 1: Identify Common Factors
  The terms $\frac{xy}{8}$, $\frac{xy^2}{16}$, and $\frac{xy^3}{20}$ have $xy$ as a common factor in the numerators.

• Step 2: Factor Out Common Numerator
  Factor $xy$ out of each term:
  $\frac{xy}{8} = \frac{xy}{8}$,
  $\frac{xy^2}{16} = \frac{xy \cdot y}{16}$,
  $\frac{xy^3}{20} = \frac{xy \cdot y^2}{20}$.

• Step 3: Simplify the Expression
  Factor out $xy$ and adjust fractions:
  $xy \left( \frac{1}{8} + \frac{y}{16} + \frac{y^2}{20} \right)$.

• Step 4: Simplify Denominator Terms
  Factor $\frac{1}{4}$ common to $\frac{1}{8}$, $\frac{y}{16}$, and $\frac{y^2}{20}$:
  $\frac{xy}{4} \left( \frac{1}{2} + \frac{y}{4} + \frac{y^2}{5} \right)$.

Thus, the expression $\frac{xy}{8} + \frac{xy^2}{16} + \frac{xy^3}{20}$ decomposes to the factored form $\frac{xy}{4} \left( \frac{1}{2} + \frac{y}{4} + \frac{y^2}{5} \right)$.

To solve the expression $16xa^2 + \frac{80x}{a} - 40a^3$ by decomposing it into factors, we need to follow a series of detailed steps:

Step 1: Identify common factors among the terms.
Observe that the terms are $16xa^2$, $\frac{80x}{a}$, and $-40a^3$.
- Each term involves either $x$ or $a$. The coefficient common factor is 8.

Step 2: Extract the common factor.
Examine each term for their factors.
- $16xa^2 = 8 \cdot 2xa^2$
- $\frac{80x}{a} = 8 \cdot \frac{10x}{a}$
- $-40a^3 = 8 \cdot (-5a^3)$
The common factor across all three is $8$.

Step 3: Factor out the common factor.
We factor out the common 8, resulting in:
$8(2xa^2 + \frac{10x}{a} - 5a^3)$.

Step 4: Further observe terms to simplify.
Notice that within $8(...)$, each term can factor out one more common factor, which is $a$:
- $2xa^2 = a \cdot 2ax$
- $\frac{10x}{a} = a \cdot \frac{10x}{a^2}$
- $-5a^3 = a \cdot (-5a^2)$
Thus, we factor out $a$, yielding:

Thus, the expression simplifies as:
$8a(2ax + \frac{10x}{a^2} - 5a^2)$.

Therefore, the decomposed form of the expression is $8a(2ax+\frac{10x}{a^2}-5a^2)$, matching choice 1.

To solve this problem, we'll proceed with the following steps:

• Step 1: Identify the greatest common factor in the expression.
• Step 2: Factor out this common factor from each term.
• Step 3: Simplify the resulting expression inside the parentheses.

Now, let's work through each step:

Step 1: The expression given is
$\frac{16xy}{z}+\frac{40x}{yz}-\frac{56x^2}{yz^2}$.
First, identify the greatest common factor (GCF) from the numerators and the highest common power in the denominators:

• The coefficients of the terms are $16$, $40$, and $56$, with a GCF of $8$.
• The common variable in the numerators is $x$.
• The shared parts of the denominators are $z$.

Thus, the GCF for the entire expression with terms considered is:

$\frac{8x}{z}$.

Step 2: Factor out $\frac{8x}{z}$ from the expression:

$\frac{8x}{z} \left( \frac{16xy}{8x}-\frac{40x}{8x}y+\frac{56x^{2}}{8x}y^{2} \right) \equiv \frac{8x}{z}(2y + \frac{5}{y} - \frac{7x}{yz})$.

Step 3: The expression in the parentheses should now be simplified:
After factoring, the expression inside the parentheses becomes $2y+\frac{5}{y}-\frac{7x}{yz}$.

Therefore, the factored form of the original expression is:

$\frac{8x}{z}(2y+\frac{5}{y}-\frac{7x}{yz})$.

To solve the problem of decomposing the expression $21ab - \frac{63a^2}{b} - 14ba^2$ into factors, we will follow these steps:

• Step 1: Identify the Greatest Common Factor (GCF).
• Step 2: Factor out the GCF.
• Step 3: Verify the result by expanding to check that it matches the original expression.

Let's go through each step:

Step 1: Find the GCF of the terms. Examine each term in the expression:

- The three terms are $21ab$, $-\frac{63a^2}{b}$, and $-14ba^2$.
- The coefficients $21$, $\frac{63}{1}$, and $14$ share a common factor of $7$.
- The variables $a$ and $b$ are present in each term. Each term has at least one $a$ and one $b$.
- Thus, the GCF is $7ab$.

Step 2: Factor out the GCF $7ab$ from the expression:

$21ab = 7ab \times 3$
$-\frac{63a^2}{b} = 7ab \times \left(-\frac{9a}{b^2}\right)$
$-14ba^2 = 7ab \times (-2a)$

Combining these, the expression factors as:

$7ab(3 - \frac{9a}{b^2} - 2a)$

Step 3: Verify by expanding the factored expression:

- Expanding $7ab(3 - \frac{9a}{b^2} - 2a)$:
$7ab \times 3 = 21ab$
$7ab \times \left(-\frac{9a}{b^2}\right) = -\frac{63a^2}{b}$
$7ab \times (-2a) = -14ba^2$

These match the original expression, confirming the factorization is correct.

Therefore, the solution to the problem is $7ab(3-\frac{9a}{b^2}-2a)$.

To solve the problem of factoring the expression $2x^3 + 4y^4 + 8z^5$, we will identify and factor out the greatest common factor (GCF) from the terms. This process involves the following steps:

• Identify the coefficients of each term: The coefficients are 2 for $2x^3$, 4 for $4y^4$, and 8 for $8z^5$.
• Determine the GCF of the numbers 2, 4, and 8. The greatest common factor is 2.
• Factor out the GCF from each term in the expression:

Let's apply this step:

Initially, our expression is $2x^3 + 4y^4 + 8z^5$.

Factoring out the GCF of 2, we rewrite each term:

• $2x^3 = 2 \cdot x^3$
• $4y^4 = 2 \cdot 2y^4$
• $8z^5 = 2 \cdot 4z^5$

Thus, the expression becomes:

$2(x^3 + 2y^4 + 4z^5)$

We have successfully factored the expression by pulling out the GCF, 2, resulting in $2(x^3 + 2y^4 + 4z^5)$.

Therefore, the final factorized expression is $2(x^3 + 2y^4 + 4z^5)$.