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Algebraic Expressions


    
1. What is an Algebraic Expression?

    
Answer: An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators (such as +, −, ×, ÷). It represents a value and does not have an equality sign. An example of an algebraic expression is:

    
3x + 5

    
In this expression, 3 is the coefficient, x is the variable, and 5 is the constant. The expression indicates a relationship between the variable and the constant, with a specific operation (addition) applied between them.


    
2. Components of an Algebraic Expression

    
Answer: An algebraic expression is made up of the following components:

    
    
        
  • Variable: A letter or symbol that represents an unknown number (e.g., x, y, a). In the expression 3x + 5, x is the variable.
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  • Coefficient: A numerical factor that multiplies the variable (e.g., in 3x + 5, 3 is the coefficient of x).
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  • Constant: A fixed value that does not change (e.g., in 3x + 5, 5 is the constant).
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  • Operators: Mathematical symbols that represent operations, such as addition (+), subtraction (−), multiplication (×), and division (÷). These operations define the relationships between the terms of the expression.
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3. Types of Algebraic Expressions

    
Answer: There are several types of algebraic expressions based on the number of terms and the powers of the variable:

    
    
        
  • Monomial: An expression with only one term. For example, 3x or 5y².
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  • Binomial: An expression with two terms. For example, 3x + 5 or 2a - 7.
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  • Trinomial: An expression with three terms. For example, x² + 2x + 3 or 2a + 3b - 7.
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  • Polynomial: An expression with more than three terms. For example, 4x³ + 3x² + 2x + 1.
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4. Simplifying Algebraic Expressions

    
Answer: Simplifying an algebraic expression means combining like terms and performing operations to make the expression as simple as possible. Here are the steps for simplifying:

    
    
        
  1. Identify like terms—terms with the same variable raised to the same power.
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  2. Combine the like terms by adding or subtracting their coefficients.
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  3. If necessary, apply distributive property to simplify further.
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For example, to simplify 3x + 5x - 2x + 7, combine like terms:

    
3x + 5x - 2x = 6x

    
The simplified expression is 6x + 7.


    
5. Evaluating Algebraic Expressions

    
Answer: To evaluate an algebraic expression, substitute a specific value for the variable and then perform the operations. For example, to evaluate the expression 3x + 5 when x = 4, follow these steps:

    
    
        
  1. Substitute the value of x into the expression: 3(4) + 5.
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  2. Perform the multiplication: 12 + 5 = 17.
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The value of the expression when x = 4 is 17.


    
6. Operations with Algebraic Expressions

    
Answer: You can perform various operations on algebraic expressions:

    
    
        
  • Addition: Combine like terms. For example, (3x + 5) + (2x - 3) = 5x + 2.
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  • Subtraction: Subtract corresponding terms. For example, (5x + 4) - (3x - 2) = 2x + 6.
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  • Multiplication: Multiply each term in one expression by each term in the other expression. For example, (x + 2)(x - 3) = x² - 3x + 2x - 6 = x² - x - 6.
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  • Division: Divide terms or use polynomial long division when dividing a polynomial by another polynomial. For example, (x² + 2x + 1) ÷ (x + 1) = x + 1.
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7. Like and Unlike Terms

    
Answer: Terms that have the same variable raised to the same power are called like terms, and they can be combined. For example, in the expression 3x + 4x, 3x and 4x are like terms, and their sum is 7x.

    
Terms with different variables or powers are called unlike terms and cannot be combined. For example, 3x and 4y are unlike terms because they have different variables.


    
8. The Distributive Property

    
Answer: The distributive property states that for any numbers a, b, and c:

    
a(b + c) = ab + ac

    
This property allows you to multiply a term outside the parentheses by each term inside the parentheses. For example:

    
3(x + 4) = 3x + 12


    
9. Factorizing Algebraic Expressions

    
Answer: Factorizing an algebraic expression means breaking it down into a product of simpler expressions. For example, factorizing the expression 6x + 9:

    
First, find the greatest common factor (GCF), which is 3:

    
6x + 9 = 3(2x + 3)

    
The expression 6x + 9 is now factorized into 3(2x + 3).


    
10. Zero Product Property

    
Answer: The Zero Product Property states that if the product of two factors is zero, at least one of the factors must be zero. In other words:

    
ab = 0 implies a = 0 or b = 0.

    
For example, if (x - 3)(x + 5) = 0, then either x - 3 = 0 or x + 5 = 0, giving the solutions x = 3 and x = -5.



Algebraic Expressions: Questions and Answers


    
1. Question: Simplify the algebraic expression 4x + 7x - 3y + 2y.

    
Answer: Combine like terms. The like terms are 4x and 7x (both have the variable x) and -3y and 2y (both have the variable y).

    
4x + 7x = 11x

    
-3y + 2y = -y

    
The simplified expression is: 11x - y.


    
2. Question: Evaluate the expression 5x² - 3x + 4 when x = 2.

    
Answer: To evaluate the expression, substitute x = 2 into the expression 5x² - 3x + 4:

    
5(2)² - 3(2) + 4

    
5(4) - 6 + 4 = 20 - 6 + 4 = 18

    
The value of the expression when x = 2 is 18.


    
3. Question: Factor the expression x² - 5x + 6.

    
Answer: We need to find two numbers that multiply to 6 and add up to -5. These numbers are -2 and -3. Therefore, we can factor the expression as:

    
x² - 5x + 6 = (x - 2)(x - 3)

    
The factored form of the expression is (x - 2)(x - 3).


    
4. Question: Expand and simplify (x + 3)(x - 4).

    
Answer: Use the distributive property (FOIL method) to expand the expression:

    
(x + 3)(x - 4) = x(x - 4) + 3(x - 4)

    
x(x - 4) = x² - 4x

    
3(x - 4) = 3x - 12

    
Now, combine like terms:

    
x² - 4x + 3x - 12 = x² - x - 12

    
The expanded and simplified expression is x² - x - 12.


    
5. Question: What is the greatest common factor (GCF) of the expression 6x² + 9x?

    
Answer: To find the GCF, we first look at the coefficients of the terms (6 and 9). The GCF of 6 and 9 is 3.

    
Next, both terms contain x, so the GCF also includes x.

    
The GCF of 6x² + 9x is 3x.

    
Now, factor out the GCF:

    
6x² + 9x = 3x(2x + 3)

    
The factored form is 3x(2x + 3).


    
6. Question: Simplify the expression 7x + 3 - 4x + 8.

    
Answer: Combine like terms:

    
7x - 4x = 3x and 3 + 8 = 11.

    
The simplified expression is: 3x + 11.


    
7. Question: Factor the expression 4x² - 12x.

    
Answer: To factor 4x² - 12x, first identify the greatest common factor (GCF). The GCF of 4x² and -12x is 4x.

    
Now, factor out the GCF:

    
4x² - 12x = 4x(x - 3)

    
The factored form is 4x(x - 3).


    
8. Question: Simplify the expression (2x + 5)(3x - 2).

    
Answer: Use the distributive property (FOIL method) to expand the expression:

    
(2x + 5)(3x - 2) = 2x(3x - 2) + 5(3x - 2)

    
2x(3x - 2) = 6x² - 4x

    
5(3x - 2) = 15x - 10

    
Now, combine all the terms:

    
6x² - 4x + 15x - 10 = 6x² + 11x - 10

    
The simplified expression is 6x² + 11x - 10.


    
9. Question: What is the value of 3a + 2b when a = 4 and b = -2?

    
Answer: To evaluate the expression, substitute the values of a and b:

    
3(4) + 2(-2) = 12 - 4 = 8

    
The value of the expression when a = 4 and b = -2 is 8.


    
10. Question: Factor the quadratic expression x² - 7x + 10.

    
Answer: We need to find two numbers that multiply to 10 and add up to -7. These numbers are -5 and -2. Therefore, we can factor the expression as:

    
x² - 7x + 10 = (x - 5)(x - 2)

    
The factored form is (x - 5)(x - 2).