Multiplication of Algebraic Expressions

Multiplication of Algebraic Expressions
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Olivia recently learned about algebraic expressions and now she is wondering if she can multiply those expressions in the same way as she multiplies the two whole numbers or fractions. 

multiplication of algebraic expressions

In this mini-lesson, let's help Olivia with the multiplication of algebraic expressions

Lesson Plan

What Do You Mean by Like and Unlike Terms?

Like Terms 

In Algebra, the terms that contain the same variable(s) which is(are) raised to the same power(s) are called like terms. 

In the like terms, the numerical coefficients can vary.

We add/subtract the like terms to simplify an expression.

For example,

Consider \(2y + 12y\)

Here, \(2y\) and \(12y\) are like terms.

Hence, they can be added directly by adding their coefficients. i.e., \(2+12=14\)

Thus, the simplified form of the given expression is \(14y\) 

Similarly, we can perform all the subtraction on the like terms. 

Unlike Terms 

In Algebra, the terms that are not like terms are called the, unlike terms. 

For example,

Consider \(3y + 5x\) is unlike terms.

 \(3y + 5x\) is unlike terms.

Hence, for further simplification, this algebraic expression can not be added directly because these are the unlike terms (as the terms have different variables \(x\) and \(y\)).


How Do You Multiply Algebraic Expressions?

In the multiplication of algebraic expressions, first, let's recall two simple rules. 

(i) The product of two terms with the same signs is positive, and the product of two terms with different signs is negative.

For example:

  • \(-a \times -\text {b is ab} \)
  • \(a \times -\text {b is -ab} \)

(ii) We can add the exponents of the same bases in case of multiplication, then

\[x^m  \times x^n = x^{m+n}\]

For example:

  • \((x^2 \times x^3) = x^5\)
  • \((x^5 \times x^8) = x^{13}\) 

Few methods for multiplying algebraic expressions

I. Multiplication of Two Monomials

Product of two monomials = numerical coefficients \(\times\) variable parts

Example

Find the product of  \(6xy \text{ and} -3x^{2}y^{3}\).

Solution

\[\begin{align*} &6xy \times -3x^{2}y^{3}\\[0.2cm]&= {6 \times -3} \times {xy \times x^2y^3}\\[0.2cm] &= -18x^{1+2}y^{1+3}\\[0.2cm]&= -18x^3y^4 \end{align*}\] 

II. Multiplication of a Polynomial by a Monomial

Multiply each term of the polynomial by the monomial, using the distributive law: \[a \times (b + c) = a \times b + a \times c\]

Example 

Find each of the following product:

\[5a^2b^2 × (3a^2 - 4ab + 6b^2)\] 

Solution

\[\begin{align*}&5a^2b^2 × (3a^2 - 4ab + 6b^2)= \\ &= (5a^2b^2) × (3a^2) + (5a^2b^2) × (-4ab) + (5a^2b^2) × (6b^2) \\ &= 15a^4b^2 - 20a^3b^3 + 30a^2b^4 \end{align*}\]

III. Multiplication of Two Binomials

We multiply two binomials by using the distributive law of multiplication twice.

Let us find the product of two binomials \((a + b)\) and \((c + d)\).

\[\begin{align*} &(a + b) \times (c + d) =\\ &= a \times (c + d) + b \times (c + d) \\ &= a \times   c + a \times d+ b \times c + b \times d \\ &= ac + ad + bc + bd \end{align*}\]

Note: This method is known as the horizontal multiplication method. 

Example 

Multiply \((3x + 5y)\) and \((5x - 7y)\).

Solution

(I) Horizontal multiplication method

\[\begin{align*}&(3x + 5y) \times (5x - 7y) =\\ &= 3x \times (5x - 7y) + 5y\times (5x - 7y) \\ &= (3x \times 5x - 3x \times 7y) + (5y \times 5x - 5y\times 7y) \\ &= (15x^2 - 21xy) + (25xy - 35y^2) \\ &= 15x^2 - 21xy + 25xy - 35y^2 \\ &= 15x^2 + 4xy - 35y^2.\end{align*}\]

(II) Column wise multiplication

    \(3x + 5y\)

\(\times (5x - 7y)\)
_____________
  \(15x^2 + 25xy\)                  ⇐ multiplication by 5x.
          \( - 21xy - 35y^2\)       ⇐ multiplication by -7y.
__________________
\(15x^2  +  4xy  - 35y^2\)       ⇐ Added the above terms.
__________________

IV. Multiplication by Polynomial

Example

Multiply \((5x^2 – 6x + 9)\) with \((2x -3)\)

Solution 

   \( 5x^2 – 6x + 9 \)

\(\times    (2x - 3)\)
____________________
   \(10x^3 - 12x^2 + 18x \)              ⇐ multiplication by 2x.

           \( - 15x^2 + 18x - 27\)        ⇐ multiplication by -3
______________________
 \(10x^3 – 27x^2 + 36x - 27 \)        ⇐ Added the above terms.
______________________

\(\therefore (5x^2 – 6x + 9) \times (2x - 3) \text{ is }(10x^3 – 27x^2 + 36x – 27)\)

 
important notes to remember
Important Notes
  1. We can add or subtract terms only if they are like terms. 
  2. For multiplication or division, the terms can be either like terms or unlike terms.  

  3. Multiplication of algebraic expression can be done by using algebraic identities.  

  4. Remember to distribute the negative sign also to all terms in the bracket.

Solved Examples

Example 1

 

 

Sam is asked to identify the like terms in the algebraic expression \(3x^2y + 6xy – xy^2 – 6x^2y\). Can we help him? 

Solution

Here, the like terms are \(3x^2y\) and \( – 6x^2y\) as each term has the same literal coefficients \(x^2y\).

\(\therefore \text{ The like terms are }3x^2y \text{ and }– 6x^2y\)

Example 2

 

 

Multiply the following polynomial by the given monomial term, using the distributive law \(a \times (b + c) = a \times b + a \times c\).

\[(-3x^2y) × (4x^2y - 3xy^2 + 4x - 5y)\]

Solution

\[ \begin{align*}&(-3x^2y) \times (4x^2y - 3xy^2 + 4x - 5y) = \\ &=(-3x^2y)\times(4x^2y)+(-3x^2y)\times(-3xy^2)+(-3x^2y)\times(4x)+(-3x^2y)\times(-5y)\\ &= -12x^4y^4 + 9x^3y^3 - 12x^3y +15x^2y^2 \end{align*}\]

Therefore, the required product is:

\(-12x^4y^4 + 9x^3y^3 - 12x^3y + 15x^2y^2\)

Example 3

 

 

Can we help Mia find the product of \((3x^2 + y^2)\) and \( (2x^2+ 3y^2)\) by using the distributive law?

Solution

\[\begin{align*}&(3x^2+y^2)(2x^2+3y^2)= \\ &= 3x^2 (2x^2 + 3y^2) + y^2 (2x^2 + 3y^2) \\ &= (6x^4 + 9x^2y^2) + (2x^2y^2 + 3y^4) \\ &= 6x^4 + 9x^2y^2 + 2x^2y^2 + 3y^4 \\ &= 6x^4 + 11x^2y^2 + 3y^4\end{align*}\]

Therefore, the required product is:

\( 6x^4 + 11x^2y^2 + 3y^4\)
 
Challenge your math skills
Challenging Questions
 
  1. Multiply \((2x³ – 5x² – x + 7)\) by \((3 - 2x + 4x²)\)
  2. Find the volume of a cube whose length is 6px units, breadth is \(5qy\) units, and height is \(2rz\) units. 

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 multiplication of algebraic expressions. The math journey around multiplication of algebraic expressions starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. You can find multiplying polynomials worksheets and multiplication of algebraic expressions worksheets at the bottom of this page. 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!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


FAQs on Multiplication of Algebraic Expressions

1. What are the types of algebraic expressions?

There are mainly three types of algebraic expressions which include:

  • Monomial Expression
  • Binomial Expression
  • Polynomial Expression.

2. What must be included in an algebraic expression?

The algebraic expression must include terms connected by the operator(s). A term is a combination of coefficient, variable(s), and/or constant(s).

3. How do you simplify algebraic expressions?

The basic steps to simplify an algebraic expression:

1. Remove parentheses by using the distributive property.
2. Use the exponent rules to multiply the terms with the same bases.
3. Combine the like terms.

4. How do you multiply algebraic expressions?

Multiplication of algebraic expressions is done by using algebraic identities. Few algebraic identities are given below: 

\[\begin{align} &(a + b)^2 =  a^2 + 2ab + b^2 \\ &(a - b)^2  =  a^2 - 2ab + b^2  \\ &(a + b)(a − b) = a^2 − b^2  \\ &(x\!+\! a) (x\!+\! b) = x^2 \!+\! x(a\!+\!b) \!+\! ab \end{align}\]

5. Do you multiply like terms?

Yes, we can multiply like terms.

We can multiply unlike terms also.

6. How do you multiply, unlike terms?

To multiply unlike terms:

  • We multiply the numerical coefficients.
  • We then multiply the variables.

7. How do you multiply three algebraic expressions?

To multiply three algebraic expressions:

  • We first multiply any two algebraic expressions.
  • We then multiply this product by the third algebraic expression.

8. How do you simplify algebraic expressions without brackets?

In this case, we just combine the like terms.

9. How do you simplify algebraic expressions with brackets?

The basic steps to simplify an algebraic expression:

1. Remove parentheses by using the distributive property.
2. Use the exponent rules to multiply the terms with the same bases.
3. Combine the like terms.

10. How to multiply rational algebraic expressions?

To multiply rational algebraic expressions few steps are given below:

  • Completely factor all numerators and denominators.
  • Cancel all common factors.
  • Either multiply the resultant fractions or leave the answer in factored form.
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