Decompose the following expression into factors:$$ 37a+6b $$

No es posible descomponer los factores de la expresión dada

Decompose the following expression into factors:$$ 20ab-4ac $$

$$ 20ab(1-\frac{1}{5}c) $$

Decompose the following expression into factors:$$ 36mn-60m $$

$$ 36mn(1-\frac{24}{n}) $$

Algebraic Method | Tutorela

It is a general term for various tools and techniques that will help us solve more complex exercises in the future.

Exponents and Powers

Powers are a shorthand way of writing the multiplication of a number by itself several times.

For example:

$4^5=4\times4\times4\times4\times4$

$4$ is the number that is multiplied by itself. It is called the "Base of power".
$5$ represents the number of times the base is multiplied by itself and it is called the "Exponent".

Detailed explanation

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Test yourself on factorization - common factor!

Which of the expressions is equivalent to the expression?

$$ 16-4c $$

Practice more now

Distributive Property

This property helps us to clear parentheses and assists us with more complex calculations. Let's remember how it works. Generally, we will write it like this:

$Z\times(X+Y)=ZX+ZY$

$Z\times(X-Y)=ZX-ZY$

Factoring: This involves taking out the common factor from within the parentheses.

The factoring method is very important. It will help us move from an expression with several terms to one that includes only one.
For example:
$2A + 4B$

This expression consists of two terms. We can factor it by reducin by the greatest common factor. In this case, it's the $2$ .
We will write it as follows:

$2A+4B=2\times(A+2B)$

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Test your knowledge

Question 1

Find the common factor:

$$ 2ax+3x $$

Question 2

Find the common factor:

$$ ab+bc $$

Question 3

Find the common factor:

$$ 7a+14b $$

Extended Distributive Property

The extended distributive property is very similar to the distributive property, but it allows us to solve exercises with expressions in parentheses that are multiplied by other expressions in parentheses.
It looks like this:

$(a+b)\times(c+d)=ac+ad+bc+bd$

In this article, we’ll explain each of these topics in detail.

In this article, we will discuss important topics within algebraic methodology. Each of these topics will be explained in more detail in their respective articles.

Reiteration: Powers

Let's return to the essential points within the topic of exponents:

In fact, exponents are a shorthand way of writing the multiplication of a number by itself several times. It looks like this:
$4^5$

$4$ is the number that is multiplied by itself. It is called the Base of the exponent.
$5$ represents the number of times the multiplication of the base is repeated and it is called the Exponent.

That is, in our example:
$4^5=4\times4\times4\times4\times4$

Let's remember that any number raised to the power of $1$ equals the number itself
That is:

$4^1=4$

And remember that any number raised to the power of $0$ equals $1$
$4^0=1$

Mathematical definition to the power of $0$ .

An important point to note is the difference between an exponent inside brackets and an exponent outside brackets. For example, what is the difference between

$(-4)^2$ and $-4^2$
It is an important case that could be confusing. When the exponent is outside of the brackets, as in the first case, you have to raise the entire expression to the given exponent, that is

$(-4)^2=(-4)\times(-4)=16$

Conversely, in the second case, one must first calculate the exponent and then deal with the negative sign. That is:

$-4^2=-(4\times4)=-16$

Also remember that exponents come before four of the operations in the order of mathematical operations, but not before parentheses.

For example:
$3\times(4-2)^2=3\times(2)^2=3\times4=12$

```

Do you know what the answer is?

Question 1

Find the biggest common factor:

$$ 12x+16y $$

Question 2

Decompose the following expression into its factors:

$$ 26a+65bc $$

Question 3

Decompose the following expression into factors:

$$ 37a+6b $$

The Distributive Property

We usually encounter the distributive property around the age of $12$ . This property is useful for clearing parentheses and assists with more complex calculations. Let's remember how it works. Generally, we write it as:

$Z \times (X + Y) = ZX + ZY$

$Z \times (X - Y) = ZX - ZY$

Now let’s look at some examples with numbers to understand the formula.

Example 1: Distributive Property

$6\times26=6\times(20+6)=6\times20+6\times6=120+36=156$

We used the distributive property to solve a problem that would have been more difficult to compute directly.
We can also use the distributive property with division operations.

Check your understanding

Question 1

Decompose the following expression into factors:

$$ 20ab-4ac $$

Question 2

Decompose the following expression into factors:

$$ 36mn-60m $$

Question 3

Decompose the following expression into factors:

$$ 13abcd+26ab $$

Example 2: Distributive Property

$104:4=(100+4):4= 100:4 + 4:4 = 25+1 = 26$

Once again, the distributive property has helped us to simplify a problem that, if solved step by step in a straightforward manner, would have been slightly more complex.

Example 3: Distributive Property with Variables

Clear the parentheses by applying the distributive property.
$3a\times(2b+5)=$

We will pay close attention to multiplying the term outside the parentheses by each of the terms inside the parentheses according to the correct order of operations.

Example 3- Distributive property with variables

Do you think you will be able to solve it?

Question 1

Which of the expressions is equivalent to the expression?

$$ 8x^2+4x $$

Question 2

Choose the expression that is equivalent to the following:

$$ 15z^2+50zx $$

Question 3

Which expression is equivalent in value to the following:

$$ 99ab^2+81b $$

Factoring: Taking Out the Common Factor from Parentheses

The method of eliminating a common factor is very important. It will help us move from an expression with several terms to one that includes only one.
For example, let's look at the expression:

$2A + 4B$

This expression is now composed of two terms. We can factorize it by eliminating the greatest common term. In this case, it's the number $2$ .
We will write it as follows:

$2A+4B=2\times(A+2B)$

We will realize that we have moved from a situation in which we had two parts being added together, to a situation with multiplication. This procedure is called factorization.
We can use the distributive property we mentioned earlier to do the reverse process. Multiply the $2$ by each of the terms inside the parentheses:

Factorization - Extracting the common term outside of the parentheses

In certain cases we might prefer an expression with multiplication, and in other cases one with addition.
In the article that elaborates on this topic, you can see more examples regarding this.

Extended Distributive Property

$(a+b)\times(c+d)=ac+ad+bc+bd$

How does the extended distributive property work?

Step 1: Multiply the first term of the first parentheses by each of the terms in the second parentheses.
Step 2: Multiply the second term of the first parentheses by each of the terms in the second parentheses.
Step 3: Combine like terms.

Example:

$(a+2)\times(3+a)=$

Test your knowledge

Question 1

Find the common factor:

$$ 2ax+4x^2 $$

Question 2

Find the common factor:

$$ 25y-100xy^2 $$

Question 3

Which of the expressions is equivalent to the expression?

$$ 16-4c $$

Phase 1: Let's multiply a by each of the terms in the second set of parentheses.

Phase 1- Let's multiply a by each of the terms inside the second parentheses

Phase 2: Let's multiply the 2 by each of the terms in the second parentheses.

Phase 2 - Multiply 2 by each of the terms in the second parentheses

Do you know what the answer is?

Question 1

Find the common factor:

$$ 2ax+3x $$

Question 2

Find the common factor:

$$ ab+bc $$

Question 3

Find the common factor:

$$ 7a+14b $$

Phase 3: Let's organize the terms and, if there are similar ones, let's associate them.

$(a+2)\times(3+a)=3a+a^2+6+2a=a^2+5a+6$

In the full article about the extended distributive property, you can find detailed explanations and many more examples.

Examples and exercises with solutions for the Algebraic Method

examples.example_title

Find the common factor:

$ab+bc$

examples.explanation_title

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

Let's consider that the common factor is b, so we will remove it:

$b(ab+bc)=$

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

$b(a+c)$

examples.solution_title

$b(a+c)$

examples.example_title

Find the common factor:

$7a+14b$

examples.explanation_title

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

We extract the common factor 7:

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

examples.solution_title

$7(a+2b)$

examples.example_title

Find the biggest common factor:

$12x+16y$

examples.explanation_title

We break down the coefficients 12 and 16 into multiplication exercises with a multiplier factor to then simplify:

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

We extract 4 which is the common factor:

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

examples.solution_title

$4(3x+4y)$

examples.example_title

Decompose the following expression into factors:

$20ab-4ac$

examples.explanation_title

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

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

examples.solution_title

$4a(5b-c)$

examples.example_title

Find the common factor:

$25y-100xy^2$

examples.explanation_title

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:

$25y(1-4xy)$

examples.solution_title

$25y(1-4xy)$

Check your understanding

Question 1

Find the biggest common factor:

$$ 12x+16y $$

Question 2

Decompose the following expression into its factors:

$$ 26a+65bc $$

Question 3

Decompose the following expression into factors:

$$ 37a+6b $$

Start practice