Factorization - Common Factor: Number of terms

Third powerExtracting a Common Factor from a group of numbers following decompositionIdentifying a Common Factor of a number following decompositionMatching expressions equal in valueComplete the missing numberDecomposing a Number into smaller numbers/partsApplying the formulaSolving the problem
Question 1

Decompose the following expression into factors:

\( 15a^2+10a+5 \)

Question 2

Decompose the following expression into factors by removing the common factor:

\( xyz+yzt+ztw+wtr \)

Question 3

Decompose the following expression into factors:

\( 4a+13b+58c \)

Question 4

Decompose the following expression into factors:

\( 14x^2y^3+21xy^4+70x^5y^2 \)

Question 5

Decompose the following expression into factors:

\( a^2b^2c^2-a^2b^5c^4+a^{-4}b^3c^7 \)

Examples with solutions for Factorization - Common Factor: Number of terms

Exercise #1

Decompose the following expression into factors:

15a2+10a+5 15a^2+10a+5

Video Solution

Step-by-Step Solution

Factor the given expression:

15a2+10a+5 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⋅5‾10=2⋅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⋅1↓5 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:

15a2+10a+55‾⋅3a2+5‾⋅(+2a)+5‾⋅(+1)↓5‾(3a2+2a+1) \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.

Answer

5(3a2+2a+1) 5(3a^2+2a+1)

Exercise #2

Decompose the following expression into factors by removing the common factor:

xyz+yzt+ztw+wtr xyz+yzt+ztw+wtr

Video Solution

Step-by-Step Solution

Factor the given expression:

xyz+yzt+ztw+wtr 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,yzt,ztw,wtr 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.

Answer

It is not possible to decompose the given expression into factors by extracting the common factor.

Exercise #3

Decompose the following expression into factors:

4a+13b+58c 4a+13b+58c

Video Solution

Step-by-Step Solution

Factor the given expression:

4a+13b+58c 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,b,c 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.

Answer

It is not possible to factorize the given expression by extracting the common factor.

Exercise #4

Decompose the following expression into factors:

14x2y3+21xy4+70x5y2 14x^2y^3+21xy^4+70x^5y^2

Video Solution

Answer

All answers are correct

Exercise #5

Decompose the following expression into factors:

a2b2c2−a2b5c4+a−4b3c7 a^2b^2c^2-a^2b^5c^4+a^{-4}b^3c^7

Video Solution

Answer

a2b2c2(a−b3c3+a−6bc5) a^2b^2c^2\left(a-b^3c^3+a^{-6}bc^5\right)

Question 1

Factor the following expression:

\( 2a^5+8a^6+4a^3 \)

Question 2

Factor the following expression:

\( 2x^3+4y^4+8z^5 \)

Question 3

Decompose the following expression into factors:

\( \frac{ab}{cd^2}+\frac{a^2b}{c^2d}+\frac{ab^2}{cd^3} \)

Question 4

Decompose the following expression into factors:

\( \frac{xy}{8}+\frac{xy^2}{16}+\frac{xy^3}{20} \)

Question 5

Decompose the following expression into factors:

\( 16xa^2+\frac{80x}{a}-40a^3 \)

Exercise #6

Factor the following expression:

2a5+8a6+4a3 2a^5+8a^6+4a^3

Video Solution

Answer

a3(25a2+8a3+4) a^3(25a^2+8a^3+4)

Exercise #7

Factor the following expression:

2x3+4y4+8z5 2x^3+4y^4+8z^5

Video Solution

Answer

2(x3+2y4+4z5) 2(x^3+2y^4+4z^5)

Exercise #8

Decompose the following expression into factors:

abcd2+a2bc2d+ab2cd3 \frac{ab}{cd^2}+\frac{a^2b}{c^2d}+\frac{ab^2}{cd^3}

Video Solution

Answer

abcd(1d+ac+bd2) \frac{ab}{cd}(\frac{1}{d}+\frac{a}{c}+\frac{b}{d^2})

Exercise #9

Decompose the following expression into factors:

xy8+xy216+xy320 \frac{xy}{8}+\frac{xy^2}{16}+\frac{xy^3}{20}

Video Solution

Answer

xy4(12+y4+y25) \frac{xy}{4}(\frac{1}{2}+\frac{y}{4}+\frac{y^2}{5})

Exercise #10

Decompose the following expression into factors:

16xa2+80xa−40a3 16xa^2+\frac{80x}{a}-40a^3

Video Solution

Answer

8a(2ax+10xa2−5a2) 8a(2ax+\frac{10x}{a^2}-5a^2)

Question 1

Decompose the following expression into factors:

\( \frac{16xy}{z}+\frac{40x}{yz}-\frac{56x^2}{yz^2} \)

Question 2

Decompose the following expression into factors:

\( 21ab-\frac{63a^2}{b}-14ba^2 \)

Exercise #11

Decompose the following expression into factors:

16xyz+40xyz−56x2yz2 \frac{16xy}{z}+\frac{40x}{yz}-\frac{56x^2}{yz^2}

Video Solution

Answer

8xz(2y+5y−7xyz) \frac{8x}{z}(2y+\frac{5}{y}-\frac{7x}{yz})

Exercise #12

Decompose the following expression into factors:

21ab−63a2b−14ba2 21ab-\frac{63a^2}{b}-14ba^2

Video Solution

Answer

7ab(3−9ab2−2a) 7ab(3-\frac{9a}{b^2}-2a)