How to factorize an equation quickly? Step by step guide

Factoring equations is a fundamental skill in mathematics. It is used to solve a variety of problems, including quadratic equations, cubic equations and polynomial equations. Factoring equations can also be used to simplify expressions and to find the greatest common factor (GCD) of two or more numbers.This guide will teach you how to factorize equations quickly and easily.

We will start with the basics and then move on to more advanced techniques. Whether you are a beginner or a pro, you will learn something new from this guide.

Define factorize

To factorize an equation is to rewrite it as the product of two or more simpler equations. This can be done by finding the common factors of the terms in the equation.

For example, the equation $x^{2} + 5 x + 6 = 0$ can be factored as $(x + 2) (x + 3) = 0$ . This is because the common factors of $x^{2}$ , $5 x$ , and $6$ are $x$ and $2$ .

Factoring equations can be useful for solving them. For example, to solve the equation $(x + 2) (x + 3) = 0$ , we can set each factor equal to zero and solve for $x$ . This gives us the solutions $x = - 2$ and $x = - 3$ .

Factoring equations can also be used to simplify expressions. For example, the expression $x^{2} + 5 x + 6$ can be simplified to $(x + 2) (x + 3)$ . This can make the expression easier to work with and understand.

How to factorize an equation

Factoring an equation involves expressing it as the product of two or more simpler expressions. This is a fundamental concept in algebra and can help you solve equations, simplify expressions and understand the behavior of functions. Here are the steps to factorize an equation:

Step 1: Identify Common Factors

– Look for common factors among the terms in the equation. These can be numbers, variables, or both. For example, in the equation `2x + 4`, you can factor out a `2` because it’s a common factor of both terms.

Step 2: Use Factoring Techniques

– Factoring by GCF (Greatest Common Factor):If there is a common factor among all terms in the equation, factor it out. For example, in `6x^2 + 9x`, you can factor out a `3x`, leaving you with `3x(2x + 3)`.

– Factoring Quadratic Expressions:If you have a quadratic equation in the form `ax^2 + bx + c`, you can use techniques like factoring by grouping, the quadratic formula, or completing the square to factor it.

– Factoring Special Patterns: Some equations may follow special factoring patterns like the difference of squares, perfect squares, or the sum or difference of cubes. Recognizing these patterns can simplify factoring. For example, `x^2 – 9` is a difference of squares and can be factored as `(x + 3)(x – 3)`.

Step 3: Check for Further Factoring

– After factoring, check if there are any common factors left within the resulting expressions. Continue factoring until you can no longer factor further.

Step 4: Set Each Factor Equal to Zero

– If you’re solving an equation, set each factor equal to zero and solve for the variable. The solutions to these simpler equations will be solutions to the original equation.

Here’s an example to illustrate these steps:

2x^2 – 5x – 3 = 0

Step 1: There are no common factors among the terms, so proceed to factoring techniques.

Step 2: To factor the quadratic expression 2x^2 – 5x – 3, you can use factoring by grouping or the quadratic formula. Factoring by grouping gives:

2x^2 – 6x + x – 3

Factor by grouping: 2x(x – 3) + 1(x – 3)

Now, you have a common factor of (x – 3) in both terms.

Step 3: Further factoring is not possible in this case.

Step 4: Set each factor equal to zero:

2x = 0 or x – 3 = 0

Step 5:Solve for `x`:

– From `2x = 0`, you get `x = 0`.

– From `x – 3 = 0`, you get `x = 3`.

So, the solutions to the equation 2x^2 – 5x – 3 = 0 are x = 0 and x = 3.

These steps should help you factorize and solve various types of equations in algebra. Remember to practice different equations to become proficient in this skill.

The method I propose is a practical approach based on two remarkable identities:

Let a and b be two real numbers:

Relation 1: (a+ b)² = a² + 2ab + b²

Similarly: (a – b)² = a² – 2ab + b²

(To derive these results, simply expand: (a + b)² = (a + b) * (a + b) = a*a + a*b + b*a + b*b = a² + ab + ba + b² = a² + 2ab + b² (since ab=ba; multiplication is commutative, it doesn’t matter which order you multiply them in, the result is the same)

Relation 2: (a + b)(a – b) = a²- b²

This method will obviously only work when factorization is possible! Yes, I am neither a mathematical genius nor a magician…

We are limiting ourselves to real numbers here.

Let’s take an example: x² + 6x – 7= 0

At first glance, is it factorizable or not?

Here’s how to proceed:

I notice that the beginning of my expression ‘x² + 6x’ is the beginning of (x + 3)²

Indeed, according to relation 1: (x + 3)² = x² + 6x + 9

So, I deduce: x² + 6x = (x + 3)² – 9

I substitute this into my expression, so I get: x² + 6x – 7 = (x + 3)² – 9 – 7 = (x + 3)² – 16

Now I use relation 2: I have the difference of two squares here because 16 = (√16)² = 4²: (x + 3)² – 4²

So, relation 2 gives me: (x + 3)² – 4² = (x + 3 + 4)(x + 3 – 4)= (x + 7)(x – 1)= 0

a² – b² =(a + b) (a – b)

Hence, x² + 6x – 7 = (x + 7)(x – 1)= 0

Remarks:

For those who know, this is equivalent to finding the roots of the polynomial, except here you don’t have to memorize anything, so there’s no formula to recall from memory…

If you end up with the sum of two squares in the end, you won’t be able to use relation 2, so you won’t be able to factorize.

Other examples:

1/ x²+ 2x +8= 0

x² + 2x is the beginning of (x + 1)²

Because (x + 1)² = x²+ 2x + 1

So, x² + 2x = (x + 1)² – 1

x² + 2x + 8= (x + 1)² – 1 + 8= (x + 1)² + 7= 0

Not factorizable because I end up with the sum of two terms, so I can’t use relation 2.

2/ x² + 4x + 3= 0

x² + 4x is the beginning of (x + 2)²

Because (x + 2)² = x² + 4x + 4

So, x² + 4x = (x + 2)² – 4= 0

Our expression becomes: x² + 4x + 3 = (x + 2)² – 4 + 3 = (x + 2)² -1

We have a difference of two square terms (because 1 = 1² !), so we use relation 2:

(x + 2)² – 1 = (x + 2 + 1) (x + 2 – 1) = (x + 3) (x + 1)

Hence, x² + 4x + 3 = (x + 3) (x + 1)= 0

3/ x² – 2x – 15= 0

x² – 2x is the beginning of (x – 1)²

Because (x – 1)² = x² – 2x + 1

So, x² – 2x = (x – 1)² – 1= 0

Our expression becomes: x² – 2x – 15 = (x – 1)² – 1 – 15 = (x – 1)² – 16

We have a difference of two square terms (because 16 = 4² ), so we use relation 2:

(x – 1)² – 16 = (x – 1)² – 4² = (x – 1 + 4) (x – 1 – 4) = (x + 3) (x – 5)

Hence, x² – 2x – 15 = (x + 3) (x – 5)= 0

4/ Slight difficulty if there’s a coefficient in front of the x² term:

4x² + 4x – 3= 0

4x² + 4x is the beginning of (2x + 1)²

Because (2x + 1)² = 4x² + 4x +1

So 4x² + 4x = (2x + 1)² – 1

Our expression becomes: 4x² + 4x – 3= (2x + 1)² – 1 – 3= (2x + 1)² – 4

We have a difference of two square terms (4 = 2²), so we use relation 2:

(2x + 1)² – 4 = (2x + 1)² – 2² = (2x + 1 + 2) (2x + 1 – 2) = (2x + 3) (2x – 1)

Hence, 4x² + 4x – 3 = (2x + 3) (2x – 1)= 0

How to factorise easy?Factorization method formula

The choice of method depends on the specific expression you are trying to factor. Here are some common factorization methods and formulas:

Common Factor (GCF) Factoring:

This method involves finding the greatest common factor of all the terms in an expression and factoring it out.

Formula: If the greatest common factor is GCF, then you can factor the expression as GCF * (remaining terms).

Example: Factoring 6x^2 + 9x, the GCF is 3x, so you factor as 3x(2x + 3).

Difference of Squares:

This method is applicable when you have an expression in the form of a^2 - b^2.

Formula: a^2 - b^2 = (a + b)(a - b).

Example: Factoring x^2 - 4, you use the formula to get (x + 2)(x - 2).

Perfect Square Trinomials:

This method is used when you have a trinomial that is a perfect square.

Formula: (a + b)^2 = a^2 + 2ab + b^2 and (a - b)^2 = a^2 - 2ab + b^2.

Example: Factoring x^2 + 4x + 4, you recognize it as (x + 2)^2.

Sum or Difference of Cubes:

This method is used for expressions in the form of a^3 ± b^3.

Formulas:

Sum of Cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Difference of Cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Example: Factoring x^3 + 8, you recognize it as a sum of cubes and use the formula to get (x + 2)(x^2 - 2x + 4).

Factoring by Grouping:

This method is useful for expressions with multiple terms.

You group terms and factor out common factors from each group.

Then, look for a common factor among the resulting expressions.

Example: Factoring 2x^2 - 3xy + 4x - 6y, you can group terms as (2x^2 - 3xy) + (4x - 6y), factor each group, and look for further common factors.

Quadratic Equations:

For quadratic expressions in the form of ax^2 + bx + c, you can use the quadratic formula to factor.

Example: Factoring x^2 - 5x + 6, you can use the quadratic formula to find the roots (x - 2)(x - 3).

Trial and Error:

In some cases, trial and error may be required to find factors that work. This is common for more complex expressions.

Example: Factoring 2x^2 + 7x + 3, you may need to try different combinations of factors of 2 and 3 until you find the correct factors.

How to factorize quadratic equations easy

Factoring quadratic equations can be made easier by following a systematic approach. Here are the steps to factorize quadratic equations easily:

Step 1: Identify the Type of Quadratic Expression

Before you begin factoring, identify the type of quadratic expression you’re dealing with. There are three common types:

Simple Quadratic Trinomials: These are quadratic expressions in the form ax^2 + bx + c where a, b, and c are constants, and a is not equal to 1.
Monic Quadratic Trinomials: These are quadratic expressions in the form x^2 + bx + c where b and c are constants.
Difference of Squares: These are quadratic expressions in the form a^2 - b^2, where a and b are constants.

Step 2: Factoring Simple Quadratic Trinomials

If you have a simple quadratic trinomial, follow these steps:

Multiply a and c (the leading coefficient and the constant term).
Find two numbers that multiply to a*c and add up to b (the coefficient of the middle term).

For example, if you have 6x^2 - 7x - 3, multiply 6 * (-3) = -18. Now, find two numbers that multiply to -18 and add up to -7. These numbers are -9 and 2.

Rewrite the middle term bx using these two numbers. In this case, rewrite -7x as -9x + 2x.
Now, group the terms and factor by grouping:6x^2 - 9x + 2x - 3
Factor each group separately:3x(2x - 3) + 1(2x - 3)
Factor out the common binomial factor:(2x - 3)(3x + 1)

You’ve successfully factored the quadratic expression.

Step 3: Factoring Monic Quadratic Trinomials

If you have a monic quadratic trinomial (with a equal to 1), the process is simpler:

Find two numbers that multiply to c (the constant term) and add up to b (the coefficient of the middle term).

For example, if you have x^2 - 5x + 6, you need to find two numbers that multiply to 6 and add up to -5. These numbers are -2 and -3.

Rewrite the middle term bx using these two numbers. In this case, rewrite -5x as -2x - 3x.
Now, group the terms and factor by grouping:x^2 - 2x - 3x + 6
Factor each group separately:x(x - 2) - 3(x - 2)
Factor out the common binomial factor:(x - 2)(x - 3)

You’ve successfully factored the quadratic expression.

Step 4: Factoring the Difference of Squares

If you have an expression in the form a^2 - b^2, it can be factored easily using the difference of squares formula:

(a^2 - b^2) = (a + b)(a - b)

For example, if you have x^2 - 9, it factors as (x + 3)(x - 3).

By following these systematic steps and identifying the type of quadratic expression you have, you can factorize quadratic equations more easily and accurately. Practice with various examples to improve your factorization skills.

How to factorize the trinomials easy

A trinomial is a polynomial with three terms. For example, the expression $x^{2} + 2 x + 1$ is a trinomial.

There are several ways to factor trinomials. The most common methods are:

Factoring out a common factor: If all of the terms in the trinomial have a common factor, you can factor it out. For example, the trinomial $x^{2} + 2 x + 1$ can be factored as $(x + 1) (x + 1)$ because all of the terms have a common factor of 1.

Factoring a perfect square trinomial: A perfect square trinomial is a trinomial that is the square of a binomial. For example, the trinomial $x^{2} + 4 x + 4$ is a perfect square trinomial because it is the square of the binomial $(x + 2)$ .

Using the sum-product pattern: The sum-product pattern states that a trinomial of the form $x^{2} + b x + c$ can be factored as $(x + r) (x + s)$ , where $r + s = b$ and $rs = c$ . For example, the trinomial $x^{2} + 5 x + 6$ can be factored as $(x + 2) (x + 3)$ because $2 + 3 = 5$ and $2 \times 3 = 6$ .

If you cannot factor the trinomial using any of these methods, you can use the quadratic formula. However, the quadratic formula is more difficult to use and less efficient than the other methods.

How to factorize polynomials step by step

Factoring polynomials involves expressing them as a product of simpler polynomials or monomials. Here’s a step-by-step guide for factorizing polynomials:

Step 1: Identify the Type of Polynomial

Determine the type of polynomial you’re working with. Polynomials can be categorized based on their degree (the highest power of the variable) and the number of terms.

Linear Polynomial: Degree 1 (e.g., ax + b)
Quadratic Polynomial: Degree 2 (e.g., ax^2 + bx + c)
Cubic Polynomial: Degree 3 (e.g., ax^3 + bx^2 + cx + d)
Higher-Degree Polynomial: Degree greater than 3

Step 2: Factor Out the Greatest Common Factor (GCF)

Look for the greatest common factor among all the terms in the polynomial and factor it out. This step simplifies the polynomial.

Step 3: Determine the Number of Terms

Depending on the number of terms, you will use different factorization techniques:

Two Terms: Look for special patterns like the difference of squares, sum of squares, or the difference or sum of cubes.
Three Terms: For trinomials (polynomials with three terms), factor by grouping or use techniques like trial and error, or the quadratic formula for quadratic trinomials.
Four or More Terms: Group terms and factor by grouping. This is often used for polynomials with more than three terms.

Step 4: Factor Based on Patterns (if applicable)

Difference of Squares (a^2 – b^2): Factors as (a + b)(a - b)

Sum of Squares (a^2 + b^2): Factors as (a + bi)(a - bi) where i is the imaginary unit.

Perfect Square Trinomials (a^2 + 2ab + b^2): Factors as (a + b)^2.

Sum of Cubes (a^3 + b^3) and Difference of Cubes (a^3 – b^3): Use the special factorization formulas:

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Step 5: Factor by Grouping (for trinomials and more terms)

For polynomials with more than three terms, group terms in pairs and factor out common factors from each pair. Then, factor out any common factors from the resulting pairs.

Step 6: Repeat Steps (4 and 5) as Necessary

Repeat the factoring steps until you can no longer simplify the polynomial further.

Step 7: Write the Factored Form

Write the final factored form of the polynomial as a product of simpler polynomials or monomials.

Step 8: Check Your Work

Multiply the factors to ensure that they indeed equal the original polynomial.

It’s important to practice factorization with various types of polynomials to become proficient. Recognizing patterns and applying the appropriate techniques is key to successful factorization.