Factorization of Algebraic Expressions

Factorization of Algebraic Expressions

We all are familiar with the term, 'Factors'.

A factor is a number that divides the given number without any remainder. It simply means expressing a number as a multiplication of two other numbers. Similarly, in Algebra we write the algebraic expressions as a product of their factors. The only difference here is that an algebraic expression involves numbers and variables combined with an arithmetic operation like addition or subtraction.

In this lesson, we will learn about factorization, its meaning, how to factorize algebraic expressions using various methods, algebraic expressions and identities solved examples on factoring of polynomials and some interactive questions at the end.

Factorisation of polynomials 

Lesson Plan

What Is Factorization of Algebraic Expressions?

'Factor' is a term used to express a number as a product of any two numbers. Factorization is a method of finding factors for any mathematical object, be it a number, a polynomial or any algebraic expression.

For example, the factors of 10 are 1,2,5 and 10. Similarly, an algebraic expression can also be factorized.

When the factors are multiplied they result in the original number or an expression that is factorized.

For example, consider the expression (x2+4x). It can be factorized as x(x+4)

When we multiply (x) and (x+4), we get the original expression (x2+4x)

Factoring and Expanding- algebraic expressions class 8 


How Do You Factorize Algebraic Expressions?

We know that an algebraic expression is made up of terms.  

For example, the term \(\begin{align} 7xy \end{align}\) can be factorized as \(\begin{align} 7 \times x \times y \end{align}\). This term cannot be factorized further. 

Now let us see the methods used to factorize algebraic expressions.

Method 1:        

Highest Common Factor Method

\(\begin{align} 2x^2 + 4x\end{align}\)

Follow the steps given below to find the factors of the expression.

Step 1:

\(\begin{align} 2x^2 \end{align}\)  can be factorized as \(\begin{align} 2 \times x \times x \end{align}\), and

\(\begin{align} 4x\end{align}\) can be factorized as \(\begin{align}2 \times 2 \times x \end{align}\).

Step 2:

Find the greatest common factor of the two terms. Here, we see that \(\begin{align} 2x\end{align}\) is the greatest common factor. We keep this factor outside the brackets, divide the polynomial terms by this factor and write the remaining expression inside the brackets.

Step 3:

Thus, the expression is factorized as \(\begin{align} 2x(x + 2)\end{align}\)

Method 2:

Applying Identities

This method involves formulas of algebraic expression for factorization.

Example 1

\(\begin{align} x^2 + 6x + 9\end{align}\)

We see that there are no common factors for the three terms in the expression.

However, in the expression, we see that \(\begin{align} 9\end{align}\) is a perfect square.

In this case, we seek the help of algebraic identities to factorize the expression easily.

This expression looks similar to the identity: \(\begin{align}(a+b)^2 = a^2 + 2ab +b^2\end{align}\)

Comparing the given expression to the identity, we get

\(\begin{align}a = x, b = 3\end{align}\)

Therefore, the factors are \(\begin{align}(x+3)^2\end{align}\) or \(\begin{align}(x+3) (x+3)\end{align}\)

Example 2

\(\begin{align}16x^2 - 25y^2\end{align}\)

In this expression, we see that there are no common factors.

Also, we see that this expression is of the form: \(\begin{align}a^2 - b^2 = (a+b)(a-b)\end{align}\)

Comparing the given expression to the identity, we get

\(\begin{align}a = 4x, b = 5y\end{align}\)

Therefore, the factors are \(\begin{align}(4x + 5y) (4x - 5y)\end{align}\)


List of Identities to Factorize Algebraic Expressions

Listed below are the formulas of algebraic expressions.

  • \(\begin{align}(a+b)^2 = a^2 + 2ab +b^2\end{align}\)
  • \(\begin{align}(a-b)^2 = a^2 - 2ab +b^2\end{align}\)
  • \(\begin{align}a^2 - b^2  = (a+b)(a-b)\end{align}\)
  • \(\begin{align}a^3 + b^3  = (a+b)(a^2-ab+b^2)\end{align}\)
  • \(\begin{align}a^3 - b^3  = (a-b)(a^2+ab+b^2)\end{align}\)
  • \(\begin{align}(a+b)^3  = a^3 + 3a^2b+3ab^2+b^3\end{align}\)
  • \(\begin{align}(a-b)^3  = a^3 - 3a^2b+3ab^2-b^3\end{align}\)
 
important notes to remember
Important Notes
  1. Factoring an algebraic expression means writing the expression as a product of factors.
  2. To verify whether the factors are correct or not, multiply them and check if you get the original algebraic expression.
  3. Algebraic expressions can be factorized using the common factor method, regrouping like terms together, and also by using algebraic identities.

Solved Examples

Example 1

 

 

Factorize \(\begin{align}x^2 - 10x + 25 \end{align}\)

Solution

In the given algebraic expression, we see that there are no common terms for all the three terms.

However, there is an identity that matches the formula of an algebraic expression.

It is,

 \(\begin{align}(a-b)^2 = a^2 - 2ab +b^2\end{align}\)

Comparing the identity to the equation we get, 

\(\begin{align}a = x, b = 5\end{align}\)

Therefore, the factors are \(\begin{align}(x-5)^2 or (x-5) (x-5)\end{align}\)

\(\therefore\) Factors of \(\begin{align}x^2 - 10x + 25 \end{align}\) are \(\begin{align}(x-5)^2 or (x-5) (x-5)\end{align}\)
Example 2

 

 

Factorize \(\begin{align}9y^2 - 16 \end{align}\)

Solution

In the given expression, we see that there are no common terms.

But both that terms can be expressed as squares. 

It is also similar to the formula of the algebraic expression \(\begin{align}a^2 - b^2  = (a+b)(a-b)\end{align}\)

Comparing the expression to the identity we get, 

\(\begin{align}a = 3y, b = 4\end{align}\)

Therefore, the factors are \(\begin{align}(3y + 4)(3y - 4)\end{align}\)

\(\therefore\) Factors of  \(\begin{align}9y^2 - 16 \end{align}\) are \(\begin{align}(3y + 4)(3y - 4)\end{align}\)
Example 3

 

 

Factorize \(\begin{align}5z^3 - 10z^2 \end{align}\)

Solution

In the given expression, the first term \(\begin{align}5z^3 \end{align}\) can be factorized as \(\begin{align}5 \times z \times z \times z\end{align}\) and the second term \(\begin{align}10z^2 \end{align}\) can be factorized as \(\begin{align}10 \times z \times z \end{align}\)

The common factor for both the terms is \(\begin{align}5z^2\end{align}\)

Taking out the common factor, we get the factors as \(\begin{align}5z^2 (z - 2)\end{align}\)

\(\therefore\) Factors of \(\begin{align}5z^3 - 10z^2 \end{align}\) are \(\begin{align}5z^2, (z - 2)\end{align}\)
Example 4

 

 

Factorize \(\begin{align}y^2 - 9y + 2y - 18\end{align}\)

Solution

In the given expression, there are four terms and we observe that there are no common factors in them.

However, the terms \(\begin{align} 9y \:and\: 18\end{align}\) have a common factor which is \(\begin{align} 9 \end{align}\). Similarly, the terms \(\begin{align}y^2 \:and\: 2y\end{align}\) have a common factor which is \(\begin{align}y\end{align}\).

Let us regroup the terms as follows.

\(\begin{align}y^2 + 2y - 9y - 18\end{align}\)

Taking out the common factor \(\begin{align}y\end{align}\) from the first two terms we get,

\(\begin{align}y(y+2)\end{align}\)

Taking out the common factor in the third and fourth terms we get,

\(\begin{align}-9(y+2)\end{align}\)

Combining the factors together we get,

\(\begin{align}(y-9)(y+2)\end{align}\)

\(\therefore\) Factors of \(\begin{align}y^2 - 9y + 2y - 18\end{align}\) are \(\begin{align}(y-9)(y+2)\end{align}\)
 
tips and tricks
Tips and Tricks
  1. To find the factors of a monomial is the same as finding the factors for a whole number. For example, \(\begin{align}27x^2\end{align}\) can be factorized as \(\begin{align}9 \times 3 \times x \times x\end{align}\)
  2. While factoring algebraic expressions, check if they are in the form of any identity. In this case, you can find the factors using the formula of the identity.

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of factorization of algebraic expressions and algebraic expressions and identities. Knowing about factors and the process of factorization helps in learning factoring of polynomials. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

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FAQs on Factorization of Algebraic Expressions

1. How do you learn factorization?

Factorization can be known by expressing a number or a mathematical expression as a product of any two numbers or a product of sum of numbers and variables.

2. What are the types of factorization?

Factorization can be done by finding common factors, by grouping like terms, and by using identities.

3. What is factorization of algebraic expressions?

An algebraic expression that is expressed as a product of factors which consists of variables, constants and arithmetic operators is called factorization of algebraic expressions.

4. What is the difference between a polynomial and an algebraic expression?

The difference between a polynomial and an algebraic expression is that the exponents of a polynomial can have only whole numbers whereas an algebraic expression has irrational numbers as exponents.

5. What is the difference between an algebraic expression and an equation?

An equation has an '=' sign, whereas, an expression does not have an '=' sign, although both have variables and numbers separated by arithmetic operators.

6. How do you solve algebraic expressions with multiplication?

To solve an algebraic expression with multiplication we multiply each term in the first expression with every other term in the second expression.

7. What are the three components of an algebraic expression?

Variables, constants (or) numbers and arithmetic operators are the three components of an algebraic expression.

8. How do you factorize algebraic expressions using the common factor method?

To factorize using the common factor method, we find the highest common factor and place it before the brackets. Then, divide the polynomial terms by this highest common factor to obtain the expression inside the brackets.

9. How many identities are there in algebraic expressions?

There are three standard identities in algebraic expressions.

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