How to Factor Algebraic Equations
Understand the definition of factoring when applied to single numbers., Understand that variable expressions can also be factored., Apply the distributive property of multiplication to factor algebraic equations.
Step-by-Step Guide
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Step 1: Understand the definition of factoring when applied to single numbers.
Factoring is conceptually simple, but, in practice, can prove to be challenging when applied to complex equations.
Because of this, it's easiest to approach the concept of factoring by starting with simple numbers, then move on to simple equations before finally proceeding to more advanced applications.
A given number's factors are the numbers that multiply to give that number.
For example, the factors of 12 are 1, 12, 2, 6, 3 and 4, because 1 × 12, 2 × 6, and 3 × 4 all equal
12.
Another way to think of this is that a given number's factors are the numbers by which it is evenly divisible.
Can you find all the factors of the number 60? We use the number 60 for a wide variety of purposes (minutes in an hour, seconds in a minute, etc.) because it's evenly divisible by a fairly wide range of numbers.
The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and
60. -
Step 2: Understand that variable expressions can also be factored.
Just as lone numbers can be factored, so too can variables with numeric coefficients be factored.
To do this, simply find the factors of the variable's coefficient.
Knowing how to factor variables is useful for simplifying algebraic equations that the variables are a part of.
For example, the variable 12x can be written as a product of the factors of 12 and x.
We can write 12x as 3(4x), 2(6x), etc., using whichever factors of 12 are best for our purposes.
We can even go as far as to factor 12x multiple times.
In other words, we don't have to stop with 3(4x) or 2(6x)
- we can factor 4x and 6x to give 3(2(2x) and 2(3(2x), respectively.
Obviously, these two expressions are equal. , Using your knowledge of how to factor both lone numbers and variables with coefficients, you can simplify simple algebraic equations by finding factors that the numbers and variables in an algebraic equation have in common.
Usually, to make the equation as simple as possible, we try to search for the greatest common factor.
This simplification process is possible because of the distributive property of multiplication, which states that for any numbers a, b, and c, a(b + c) = ab + ac.
Let's try an example problem.
To factor the algebraic equation 12 x + 6, first, let's try to find the greatest common factor of 12x and
6. 6 is the biggest number that divides evenly into both 12x and 6, so we can simplify the equation to 6(2x + 1).
This process also applies to equations with negatives and fractions. x/2 + 4, for instance, can be simplified to 1/2(x + 8), and
-7x +
-21 can be factored to
-7(x + 3). -
Step 3: Apply the distributive property of multiplication to factor algebraic equations.
Detailed Guide
Factoring is conceptually simple, but, in practice, can prove to be challenging when applied to complex equations.
Because of this, it's easiest to approach the concept of factoring by starting with simple numbers, then move on to simple equations before finally proceeding to more advanced applications.
A given number's factors are the numbers that multiply to give that number.
For example, the factors of 12 are 1, 12, 2, 6, 3 and 4, because 1 × 12, 2 × 6, and 3 × 4 all equal
12.
Another way to think of this is that a given number's factors are the numbers by which it is evenly divisible.
Can you find all the factors of the number 60? We use the number 60 for a wide variety of purposes (minutes in an hour, seconds in a minute, etc.) because it's evenly divisible by a fairly wide range of numbers.
The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and
60.
Just as lone numbers can be factored, so too can variables with numeric coefficients be factored.
To do this, simply find the factors of the variable's coefficient.
Knowing how to factor variables is useful for simplifying algebraic equations that the variables are a part of.
For example, the variable 12x can be written as a product of the factors of 12 and x.
We can write 12x as 3(4x), 2(6x), etc., using whichever factors of 12 are best for our purposes.
We can even go as far as to factor 12x multiple times.
In other words, we don't have to stop with 3(4x) or 2(6x)
- we can factor 4x and 6x to give 3(2(2x) and 2(3(2x), respectively.
Obviously, these two expressions are equal. , Using your knowledge of how to factor both lone numbers and variables with coefficients, you can simplify simple algebraic equations by finding factors that the numbers and variables in an algebraic equation have in common.
Usually, to make the equation as simple as possible, we try to search for the greatest common factor.
This simplification process is possible because of the distributive property of multiplication, which states that for any numbers a, b, and c, a(b + c) = ab + ac.
Let's try an example problem.
To factor the algebraic equation 12 x + 6, first, let's try to find the greatest common factor of 12x and
6. 6 is the biggest number that divides evenly into both 12x and 6, so we can simplify the equation to 6(2x + 1).
This process also applies to equations with negatives and fractions. x/2 + 4, for instance, can be simplified to 1/2(x + 8), and
-7x +
-21 can be factored to
-7(x + 3).
About the Author
Emma Gibson
Creates helpful guides on lifestyle to inspire and educate readers.
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