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

## 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.
______________________

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

### 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.

More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus
More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus