Multiplication of Algebraic Expressions

๐Ÿ†Practice variables and algebraic expressions

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: 7ร—X+13ร—Y=7X+13Y\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.

7ร—X+13ร—Y=7X+13Y

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Test yourself on variables and algebraic expressions!

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

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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

  • 8ร—X=8X 8\times X=8X
  • Xร—Y=XY X\times Y=XY
  • โˆ’3ร—(5+6)=โˆ’3(5+6) -3\times\left(5+6\right)=-3\left(5+6\right)
  • 8ร—(Xโˆ’5)=8(Xโˆ’5) 8\times\left(X-5\right)=8\left(X-5\right)
  • (5โˆ’a)ร—(3+4)=(5โˆ’a)(3+4) \left(5-a\right)\times\left(3+4\right)=(5-a)\left(3+4\right)
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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. ย 


Examples and exercises with solutions for multiplying algebraic expressions

Exercise #1

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

Video Solution

Step-by-Step Solution

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

  • Step 1: Combine Like Terms Involving x x
    The terms 3x 3x and 4x 4x are like terms because both involve the variable x x . To combine them, add their coefficients:
    3x+4x=(3+4)x=7x 3x + 4x = (3 + 4)x = 7x

  • Step 2: Combine Constant Terms
    The expression includes constant terms 7 7 and 2 2 . These can be added together to simplify:
    7+2=9 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 7x + 9

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

Reviewing the choices provided, the correct choice is:

  • Choice 2: 7x+9 7x + 9

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

Answer

7x+9 7x+9

Exercise #2

3z+19zโˆ’4z=? 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+19zโˆ’4z 3z + 19z - 4z . The coefficients are 3 3 , 19 19 , and โˆ’4 -4 .

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

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

Therefore, the simplified expression is 18z 18z .

The solution to the problem is 18z 18z .

Answer

18z 18z

Exercise #3

Are the expressions the same or not?

20x 20x

2ร—10x 2\times10x

Video Solution

Step-by-Step Solution

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

  • Step 1: Simplify the expression 2ร—10x 2 \times 10x .
  • Step 2: Compare the simplified expression with 20x 20x .

Now, let's work through each step:
Step 1: The expression 2ร—10x 2 \times 10x can be rewritten using associativity as 2ร—(10ร—x) 2 \times (10 \times x) .
Step 2: Apply the associative property of multiplication: (2ร—10)ร—x=20ร—x=20x (2 \times 10) \times x = 20 \times x = 20x .

Comparing this with the given expression, we see that both expressions are indeed the same, as they simplify to 20x 20x .

Therefore, the solution to the problem is Yes.

Answer

Yes

Exercise #4

Are the expressions the same or not?

3+3+3+3 3+3+3+3

3ร—4 3\times4

Video Solution

Step-by-Step Solution

To solve this problem, we'll analyze the expressions 3+3+3+33+3+3+3 and 3ร—43 \times 4 to determine if they are equivalent.

First, evaluate the expression 3+3+3+33+3+3+3:

  • Add the numbers: 3+3=63 + 3 = 6
  • Add again: 6+3=96 + 3 = 9
  • Add the last 33: 9+3=129 + 3 = 12

The result of 3+3+3+33+3+3+3 is 1212.

Next, evaluate the expression 3ร—43 \times 4:

  • Perform the multiplication: 3ร—4=123 \times 4 = 12

The result of 3ร—43 \times 4 is also 1212.

Since both expressions result in the same number, we conclude that

The expressions are the same.

Therefore, the correct answer is Yes.

Answer

Yes

Exercise #5

Are the expressions the same or not?

18x 18x

2+9x 2+9x

Video Solution

Step-by-Step Solution

To determine if the expressions 18x 18x and 2+9x 2 + 9x are equivalent, we'll analyze their structures.

  • 18x 18x is a linear expression with a single term involving the variable x x , and its coefficient is 18.
  • 2+9x 2 + 9x consists of two terms: a constant term 2 2 and a linear term 9x 9x with coefficient 9.

For two expressions to be equivalent, each corresponding term must be equal. Here, the expression 18x 18x has no constant term, whereas 2+9x 2 + 9x has a constant term of 2. Furthermore, the linear term coefficients differ: 18โ‰ 9 18 \neq 9 .

Therefore, the expressions 18x 18x and 2+9x 2 + 9x are not the same. They structurally differ and cannot be made equivalent just through similar values of x x .

Therefore, the solution to this problem is: No.

Answer

No

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