Multiplication of Algebraic Expressions

We already know that when encountering expressions where a number is added to or subtracted from a variable, we cannot combine them directly.
However, when we see an expression where a number multiplies a variable, we can simplify it by applying the multiplication!

Multiplying algebraic expressions is the same as multiplying conventional numbers, and therefore, the rules we apply to these will also be applied to algebraic expressions.
For example: $\green{ 7\times X}+\red{13\times Y}=\green{7X}+\red{13Y}$

Multiplying algebraic expressions involves distributing each term in one expression across all terms in another, combining like terms where possible.
This includes basic products like monomials with monomials, binomials, and more complex polynomials.
In algebraic expressions that contain variables or parentheses, it is not necessary to write the multiplication sign.

Using the multiplication of algebraic expressions enables us to simplify and even solve equations. It also opens the door to more advanced algebraic techniques, such as the Distributive Property and the FOIL Method, which help manage complex expressions and build towards solving intricate problems in algebra.

Detailed explanation

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Test yourself on variables and algebraic expressions!

$3x+4x+7+2=\text{?}$

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Remember that the rules of multiplication for algebraic expressions are the same as the multiplication rules for numbers.

Here are some examples

$8\times X=8X$
$X\times Y=XY$
$-3\times\left(5+6\right)=-3\left(5+6\right)$
$8\times\left(X-5\right)=8\left(X-5\right)$
$\left(5-a\right)\times\left(3+4\right)=(5-a)\left(3+4\right)$

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Test your knowledge

Question 1

$3z+19z-4z=\text{?}$

Question 2

$$ 11+5x-2x+8= $$

Question 3

$$ 5+0+8x-5= $$

How many exercises should I practice?

Since each student has a different learning pace, the answer to this question is individual for each one.
The important thing is that you are aware of your level, and know if you need to practice the formulas more.
Anyway, to memorize the basic formula, it is recommended to do 10 exercises of basic and medium level.

Examples and exercises with solutions for multiplying algebraic expressions

Exercise #1

$3x+4x+7+2=\text{?}$

Video Solution

Step-by-Step Solution

Let's simplify the expression $3x + 4x + 7 + 2$ step-by-step:

Step 1: Combine Like Terms Involving $x$
The terms $3x$ and $4x$ are like terms because both involve the variable $x$ . To combine them, add their coefficients:
$3x + 4x = (3 + 4)x = 7x$
Step 2: Combine Constant Terms
The expression includes constant terms $7$ and $2$ . These can be added together to simplify:
$7 + 2 = 9$
Step 3: Write the Simplified Expression
Now, combine the results from Step 1 and Step 2 to form the final simplified expression:
$7x + 9$

Therefore, the simplified expression is $7x + 9$ .

Reviewing the choices provided, the correct choice is:

Choice 2: $7x + 9$

This matches our simplified expression, confirming our solution is correct.

Answer

$7x+9$

Exercise #2

$3z+19z-4z=\text{?}$

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Step 1: Combine like terms by identifying and adding their coefficients.
Step 2: Simplify the expression.
Step 3: Verify the resulting expression with the provided choices.

Let's work through each step:

Step 1: Identify the coefficients in the expression $3z + 19z - 4z$ . The coefficients are $3$ , $19$ , and $-4$ .

Step 2: Add and subtract these coefficients: $3 + 19 - 4$ .

Step 3: Calculate: $3 + 19 = 22$ and then $22 - 4 = 18$ .

Therefore, the simplified expression is $18z$ .

The solution to the problem is $18z$ .

Answer

$18z$

Exercise #3

$11+5x-2x+8=$

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Step 1: Identify the like terms in the expression.
Step 2: Combine the constant terms.
Step 3: Combine the coefficients of $x$ .

Now, let's work through each step:
Step 1: The given expression is $11 + 5x - 2x + 8$ . There are constants (11 and 8) and terms with $x$ (5x and -2x).
Step 2: Combine the constants: $11 + 8 = 19$ .
Step 3: Combine the coefficients of $x$ : $5x - 2x = 3x$ .

After simplification, the expression becomes $19 + 3x$ .

The correct solution from the multiple-choice options is $\boxed{19 + 3x}$ .

Answer

19+3X

Exercise #4

$5+0+8x-5=$

Video Solution

Step-by-Step Solution

To simplify the expression $5 + 0 + 8x - 5$ , follow these steps:

Step 1: Identify and group like terms. In this case, there are constants (5, 0, -5) and a term with a variable (8x).
Step 2: Combine the constants: $5 + 0 - 5$ .
Step 3: Calculate: $5 - 5 = 0$ .

Now, our expression simplifies to $0 + 8x$ , which is simply $8x$ .

Therefore, the simplified expression is $8x$ .

Answer

$8X$

Exercise #5

$5+8-9+5x-4x=$

Video Solution

Step-by-Step Solution

To solve this problem, we will simplify the expression $5+8-9+5x-4x$ by separately combining the constants and the variable terms.

Step 1: Simplify the constant terms.
$5 + 8 - 9 = 4$

Step 2: Simplify the variable terms.
$5x - 4x = x$

Step 3: Combine the results from steps 1 and 2.
Thus, the simplified expression is $4 + x$ .

Therefore, the solution to the problem is $4 + x$ , which corresponds to choice .

Answer

4+X

Do you know what the answer is?

Question 1

$$ 5+8-9+5x-4x= $$

Question 2

$$ x+x= $$

Question 3

Are the expressions the same or not?

$$ 18x $$

$$ 2+9x $$

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Multiplication of Algebraic Expressions