What Is An Example Of The Distributive Property

The distributive property is a fundamental concept in algebra that allows you to simplify expressions by multiplying a single term by two or more terms inside a set of parentheses. Mastering the distributive property is crucial for success in algebra and beyond, as it provides a powerful tool for simplifying complex equations and solving for unknown variables.

Understanding the Distributive Property

At its core, the distributive property states that multiplying a number by the sum or difference of two other numbers is the same as multiplying the number by each of the other numbers individually, and then adding or subtracting the products.

Mathematically, this can be represented as:

• a(b + c) = ab + ac
• a(b - c) = ab - ac

Where 'a', 'b', and 'c' represent any real numbers.

Let's break down what this means:

• a(b + c): This means you are multiplying 'a' by the sum of 'b' and 'c'.
• ab + ac: This means you are multiplying 'a' by 'b', multiplying 'a' by 'c', and then adding the two results.

The distributive property works because multiplication is inherently a shorthand way of representing repeated addition.

Visualizing the Distributive Property

Imagine you have a rectangular garden that is 5 meters wide. One section of the garden is 3 meters long, and another section is 4 meters long. The total length of the garden is therefore 3 + 4 = 7 meters.

You can calculate the total area of the garden in two ways:

1. Calculate the total length first: The total length is 7 meters, and the width is 5 meters. So the area is 5 * 7 = 35 square meters.
2. Calculate the area of each section separately: The first section has an area of 5 * 3 = 15 square meters. The second section has an area of 5 * 4 = 20 square meters. The total area is 15 + 20 = 35 square meters.

This illustrates the distributive property: 5 * (3 + 4) = (5 * 3) + (5 * 4).

Examples of the Distributive Property in Action

Let's look at several examples of how to apply the distributive property to simplify algebraic expressions.

Example 1: Simple Numerical Distribution

Problem: Simplify 4(2 + 5)

Solution:

1. Apply the distributive property: 4(2 + 5) = (4 * 2) + (4 * 5)
2. Perform the multiplication: (4 * 2) + (4 * 5) = 8 + 20
3. Simplify: 8 + 20 = 28

Therefore, 4(2 + 5) = 28. You can verify this by directly adding 2 + 5 = 7, and then multiplying 4 * 7 = 28.

Example 2: Distribution with Variables

Problem: Simplify 3(x + 2)

Solution:

1. Apply the distributive property: 3(x + 2) = (3 * x) + (3 * 2)
2. Perform the multiplication: (3 * x) + (3 * 2) = 3x + 6

Therefore, 3(x + 2) = 3x + 6. This expression is now simplified, as 3x and 6 are not like terms and cannot be combined.

Example 3: Distribution with Subtraction

Problem: Simplify 7(y - 3)

Solution:

1. Apply the distributive property: 7(y - 3) = (7 * y) - (7 * 3)
2. Perform the multiplication: (7 * y) - (7 * 3) = 7y - 21

Therefore, 7(y - 3) = 7y - 21.

Example 4: Distribution with Negative Numbers

Problem: Simplify -2(a + 4)

Solution:

1. Apply the distributive property: -2(a + 4) = (-2 * a) + (-2 * 4)
2. Perform the multiplication: (-2 * a) + (-2 * 4) = -2a - 8

Therefore, -2(a + 4) = -2a - 8. Remember that multiplying a negative number by a positive number results in a negative number.

Example 5: Distribution with Negative Variables

Problem: Simplify 5(-b + 1)

Solution:

1. Apply the distributive property: 5(-b + 1) = (5 * -b) + (5 * 1)
2. Perform the multiplication: (5 * -b) + (5 * 1) = -5b + 5

Therefore, 5(-b + 1) = -5b + 5.

Example 6: Distribution with Coefficients

Problem: Simplify 2x(3x + 5)

Solution:

1. Apply the distributive property: 2x(3x + 5) = (2x * 3x) + (2x * 5)
2. Perform the multiplication: (2x * 3x) + (2x * 5) = 6x² + 10x

Therefore, 2x(3x + 5) = 6x² + 10x. Remember to multiply both the coefficients (numbers) and the variables. When multiplying variables with exponents, add the exponents (in this case, x * x = x^(1+1) = x²).

Example 7: Distribution with Multiple Terms

Problem: Simplify -4(2x - 3y + 1)

Solution:

1. Apply the distributive property: -4(2x - 3y + 1) = (-4 * 2x) - (-4 * 3y) + (-4 * 1)
2. Perform the multiplication: (-4 * 2x) - (-4 * 3y) + (-4 * 1) = -8x + 12y - 4

Therefore, -4(2x - 3y + 1) = -8x + 12y - 4. Remember that multiplying two negative numbers results in a positive number.

Example 8: Distribution with Fractions

Problem: Simplify (1/2)(4a - 6)

Solution:

1. Apply the distributive property: (1/2)(4a - 6) = ((1/2) * 4a) - ((1/2) * 6)
2. Perform the multiplication: ((1/2) * 4a) - ((1/2) * 6) = 2a - 3

Therefore, (1/2)(4a - 6) = 2a - 3. Multiplying by 1/2 is the same as dividing by 2.

Example 9: Combining Distribution and Like Terms

Problem: Simplify 2(x + 3) + 4x

Solution:

1. Apply the distributive property: 2(x + 3) + 4x = (2 * x) + (2 * 3) + 4x
2. Perform the multiplication: (2 * x) + (2 * 3) + 4x = 2x + 6 + 4x
3. Combine like terms: 2x + 6 + 4x = 6x + 6

Therefore, 2(x + 3) + 4x = 6x + 6. Remember to combine terms with the same variable and exponent.

Example 10: Distribution on Both Sides of an Equation

Problem: Solve for x: 3(x + 2) = 2(x + 5)

Solution:

1. Apply the distributive property on both sides: 3(x + 2) = 2(x + 5) becomes 3x + 6 = 2x + 10
2. Isolate the variable terms: Subtract 2x from both sides: 3x + 6 - 2x = 2x + 10 - 2x which simplifies to x + 6 = 10
3. Isolate the constant terms: Subtract 6 from both sides: x + 6 - 6 = 10 - 6 which simplifies to x = 4

Therefore, the solution to the equation is x = 4.

Example 11: Distribution with Exponents

Problem: Simplify x²(x³ + 2x)

Solution:

1. Apply the distributive property: x²(x³ + 2x) = (x² * x³) + (x² * 2x)
2. Perform the multiplication: (x² * x³) + (x² * 2x) = x^(2+3) + 2x^(2+1) = x⁵ + 2x³

Therefore, x²(x³ + 2x) = x⁵ + 2x³. Remember to add the exponents when multiplying variables with the same base.

Example 12: More Complex Distribution

Problem: Simplify (a + b)(c + d)

Solution:

This example requires distributing twice. Think of it as distributing 'a' and 'b' separately over (c + d).

1. Distribute 'a': a(c + d) = ac + ad
2. Distribute 'b': b(c + d) = bc + bd
3. Combine the results: ac + ad + bc + bd

Therefore, (a + b)(c + d) = ac + ad + bc + bd. This is often remembered by the acronym FOIL (First, Outer, Inner, Last).

Example 13: Distribution and Simplification with Multiple Variables

Problem: Simplify 3x(2x + y) - 2y(x - y)

Solution:

1. Distribute 3x: 3x(2x + y) = 6x² + 3xy
2. Distribute -2y: -2y(x - y) = -2xy + 2y²
3. Combine the results: 6x² + 3xy - 2xy + 2y²
4. Combine like terms: 6x² + xy + 2y²

Therefore, 3x(2x + y) - 2y(x - y) = 6x² + xy + 2y².

Example 14: Distribution with Radicals

Problem: Simplify √2 (√2 + 3)

Solution:

1. Apply the distributive property: √2 (√2 + 3) = (√2 * √2) + (√2 * 3)
2. Perform the multiplication: (√2 * √2) + (√2 * 3) = 2 + 3√2

Therefore, √2 (√2 + 3) = 2 + 3√2. Remember that √2 * √2 = 2.

Example 15: Application in Geometry

Problem: The side of a square is represented by the expression (x + 5). Find an expression for the area of the square.

Solution:

The area of a square is side * side, or side². In this case, the area is (x + 5)².

1. Rewrite the expression: (x + 5)² = (x + 5)(x + 5)
2. Apply the distributive property (FOIL): (x + 5)(x + 5) = x² + 5x + 5x + 25
3. Combine like terms: x² + 5x + 5x + 25 = x² + 10x + 25

Therefore, the area of the square is represented by the expression x² + 10x + 25.

Common Mistakes to Avoid

• Forgetting to distribute to all terms: Make sure to multiply the term outside the parentheses by every term inside.
• Incorrectly applying the sign: Pay close attention to negative signs. Remember that a negative times a negative is a positive, and a negative times a positive is a negative.
• Combining unlike terms: You can only combine terms that have the same variable and exponent. For example, you can combine 3x and 5x, but you cannot combine 3x and 5x².
• Order of operations: Remember to follow the order of operations (PEMDAS/BODMAS). Distribution should be done before addition or subtraction of terms outside the parentheses.

Importance of the Distributive Property

The distributive property is essential for several reasons:

• Simplifying expressions: It allows you to rewrite complex expressions in a more manageable form.
• Solving equations: It is a key step in solving many algebraic equations.
• Factoring: The distributive property is the basis for factoring, which is the reverse process of distribution.
• Advanced mathematics: It is a foundational concept used in calculus, linear algebra, and other advanced areas of mathematics.

Practice Problems

To solidify your understanding of the distributive property, try solving the following problems:

1. 5(a - 2)
2. -3(2b + 4)
3. x(x + 7)
4. 2y(3y - 1)
5. (p + 3)(q - 2)
6. 4(z + 1) - 2z
7. (1/3)(6c + 9)
8. -2x(x² - 5x)
9. (m - n)(m + n)
10. √3 (2√3 - 1)

(Answers will be provided below).

Conclusion

The distributive property is a fundamental tool in algebra that allows you to simplify expressions and solve equations. By understanding its principles and practicing its application, you can build a strong foundation for success in mathematics. Mastering this property will significantly enhance your problem-solving abilities and open doors to more advanced mathematical concepts. Remember to pay attention to signs, distribute to all terms within the parentheses, and combine like terms to achieve accurate results. With consistent practice, you'll become proficient in using the distributive property to confidently tackle a wide range of algebraic challenges.

Answers to Practice Problems:

1. 5a - 10
2. -6b - 12
3. x² + 7x
4. 6y² - 2y
5. pq - 2p + 3q - 6
6. 2z + 4
7. 2c + 3
8. -2x³ + 10x²
9. m² - n²
10. 6 - √3