What is factoring?
In maths, factoring is a method in which you find several expressions and multiply them together to make a given number or equation. Factoring is a fundamental method to solve algebraic equations and problems.
Factoring can be used to simplify complex equations in algebra. It helps you to eliminate specific possible answers quicker than you will be able to solve manually.
Factoring is a very simple process, but it might sound tedious sometimes. You can understand the concept of factoring by starting with simple numbers. For example, the factors for 16 are 1, 16, 2,8, and two 4s. As 1×16, 2×8, 4×4 all give you the same result, and that is 16.
Another way to understand factors is, the given number’s factors are evenly divisible by the factors.
So, what will be the factors of 80?
Well, if you solve it out, it will be 1, 2, 3, 4, 5, 8, 10, 16, 20, 40, and 80.
How to factorize variable expression
Now, we were dealing with known numbers, and it was easy to factorize it. Moving ahead, we will see how to factorize a variable expression.
Just like numbers, numeric coefficients can be factored too. If you know how to factorize the variable equation, it helps you to simplify complex algebraic equations.
For example, you can write 16x as a product of factors 16 and x. We can also write 16x as 4(4x), 2(8x), etc. using the factors of 16 that are best for our purposes. After this, we can further factorize it into 2(2(2(2x))) and 2(2(2(2(2x)))).
Applying distributive property
We all have studied the concept of distributive property in our elementary education. You can simplify algebraic equations by finding factors that the numbers and variables in the equation have in common. Usually, to make it as simple as possible, we find the Greatest Common Factor.
The distributive property states this: a(b+c) = ab +ac where a, b and c are three numbers.
Example: To factorize 21x – 7 first, we will find the greatest common factor of 21x, and 7. 7 is the biggest number that divides both 21x and 7, so we can write it as 7(3x – 1).
Equations like (x/4 + 1) are simplified to 1/4(x+4) and -7x-21 to the final answer as -7(x+3).
FactoringFactorizing a quadratic equation
You will study that a quadratic equation is in the form of ax2 + bx + c = 0.
In this equation, a, b and c are numbers, and x is the variable. Everything in this is equal to 0 on the right-hand side; hence you can shift terms to the other side.
Example: 5x2 +7x – 9 = 4x2 +x -18 is solved as x2 + 6x + 9 = 0.
Equations that have the power of more than 2 like 3, 4, etc. are not quadratic. They are cubic equations; hence this solution won’t apply to those equations.
If your equation is in the form of x2 +bx + c = 0, then you can simplify this term using a shortcut to factor the equation.
Find out 2 numbers that both multiply to make c and add to make b. Once you find these two numbers d and e, you can place them in the expression (x+d) (x+e). When you multiply these two expressions, you will get the same quadratic equation. Hence, they are the factors of your quadratic equation.
Example: If your equation is x2 +7x +10 = 0.
When you multiply 5 and 2, you get 10, and when you add 5 and 2, you get 7, so you can simplify the equation as (x+5) (x+2).
You can follow a simple template
- if your equation is in the form of x2 – bx + c then your answer is in this template (x -)(x -)
- If your equation is in the form of x2 + bx + c then your answer is in this template (x +) (x+)
- If your equation is in the form of x2 – bx – c then your answer is in this template (x +) (x-)
The numbers in the blanks can also be fractions or decimals. For example: x/4 + 1 is simplified to 1/4(x+4) and -7x -21 which equals to -7(x+3).
Factoring using squares
In some cases, quadratic equations are easily factored by completing the square. A quadratic equation in this form: x2 + 2xp + p2 = (x + p)2.
So, in the equation your x value is twice the square root of the p-value, your equation can be factored as (x + p)2.
Example: the equation x2 + 8x + 16 fits this form.
42 is 16 and 4×2 is 8 and the factors will be (x + 4)2.
This is an easy way to factorize the equation. It saves you a lot of time.
Using factors to solve quadratic equations
Now after finding equations like (x + 2) (x + 2) what do we do next to find the x?
Well, now we will see how to use these factors to search the two values of x. You can solve the value of x by setting each factor equal to 0 and find the answer. You are looking for values that cause your equation = 0, so there are two probabilities.
Example: x2 + 8x + 16 = 0
Now, you can easily find the factors of this equation as (x + 4) (x + 4).
If anyone of the factors is zero then the whole equation is 0. So, x can be -4 and -4 for (x + 4) and (x + 4) respectively.
Factoring other types of equations
There is another type of equation in algebra. They are of the form a2 – b2 and have the factors as (a + b) and (a – b). Equations with two variables have factors different from basic quadratics.
Example: 9x2 – 4y2 = (3x +2y) (3x – 2y).
If a and b are not equal to 0, then the factors you will get are in the form of (a + b) and (a – b).
Cubic equations and other higher-power equations can be difficult to factorize at times. Now, if the equation is in the form of a3 – b3, the factor becomes more complicated. It comes down to this form: (a – b) (a2 + ab + b2).
Example: 8x3 – 27y3 have factors (2x – 3y) (4x2 + ((2x) (3y)) + 9y2).
Let us take a look at another type of example of factoring in algebra which you will find commonly.
3y2 + 18y which you can also write as 3 (y2 + 6y),
But in this case, 3y2 is also 3y*y and 18y is 18*y ,
So, after factoring the original equation, you can write 3y2 + 18y as 3y (y + 6).
Formulas to remember
There are some formulas which will be useful when you solve polynomials. Let us have a look at them:
a2 – b2 = (a+b)(a−b)
a2 + 2ab + b2 = (a+b)2
a2 − 2ab + b2 = (a−b)2
a3 – b3 = (a−b)(a2+ab+b2)
a3+3a2b+3ab2+b3 = (a+b)3
a3−3a2b+3ab2−b3 = (a−b)3
Algebra can be a difficult topic for many students, but with constant practice, they can excel in this area of mathematics correctly. Solving Algebra made it easy online for teachers and parents who can take guidance from different books and resources to educate their children coherently and quickly.
There are many resources online which cater to answers only while a few give you explainer videos and notes to understand the concepts well. Solving previous year question papers is the best technique to learn algebra and apply it while you are solving questions from your textbook.
Algebra is a broad part of mathematics that includes number theory, geometry, and analysis. It is also used in coordinate geometry to solve different problems on the plane.
In history, arithmetic and geometry were the two broad divisions of mathematics but later as the subject evolved, algebra was used in both fields.
The questions that come for the students in exams are generally quadratic equations and not those that have the power of 3 or 4. The equations containing those powers are complex and difficult to solve in many cases. For classes 8 to 10, students must practice equations that have a power of 2.
After practicing a lot, they will understand the technique, and solving polynomials becomes easier. Follow FarhanTech for more!