Distributive Property Rewriting And Simplifying The Expression -4(7-2)

by ADMIN 71 views

The distributive property is a fundamental concept in algebra that allows us to simplify expressions involving multiplication and addition or subtraction. It's a powerful tool for rewriting expressions and solving equations, making it an essential skill for anyone studying mathematics. In this article, we'll delve into the distributive property, explore how to apply it, and work through an example to solidify our understanding.

At its core, the distributive property states that multiplying a single term by a sum or difference inside parentheses is the same as multiplying the term by each part of the sum or difference individually and then adding or subtracting the results. Mathematically, this can be expressed as:

  • a(b + c) = a * b + a * c
  • a(b - c) = a * b - a * c

Here, 'a' is the term being distributed, and 'b' and 'c' are the terms inside the parentheses. The property holds true for both addition and subtraction, making it a versatile tool in algebraic manipulations.

Understanding the distributive property is crucial because it simplifies complex expressions and makes them easier to work with. For instance, consider the expression 3(x + 2). Without the distributive property, we would be stuck. However, by applying the property, we can rewrite the expression as 3 * x + 3 * 2, which simplifies to 3x + 6. This simple transformation can be a game-changer when solving equations or simplifying larger expressions.

The distributive property also plays a vital role in combining like terms. Like terms are terms that have the same variable raised to the same power. For example, in the expression 2x + 3x + 4, 2x and 3x are like terms. By using the distributive property in reverse, we can factor out the common variable 'x' and combine the coefficients: 2x + 3x = x(2 + 3) = 5x. This process is essential for simplifying algebraic expressions and solving equations efficiently.

Furthermore, the distributive property extends beyond simple expressions with single variables. It can be applied to expressions with multiple variables, constants, and even more complex algebraic structures. For example, consider the expression (x + 1)(x + 2). To expand this expression, we distribute each term in the first set of parentheses over the terms in the second set:

  • (x + 1)(x + 2) = x(x + 2) + 1(x + 2)
  • = x * x + x * 2 + 1 * x + 1 * 2
  • = x^2 + 2x + x + 2
  • = x^2 + 3x + 2

This process, often referred to as the FOIL method (First, Outer, Inner, Last), is a direct application of the distributive property and is fundamental in algebra.

In summary, the distributive property is a cornerstone of algebra. It allows us to simplify expressions, combine like terms, and expand products of binomials. Mastering this property is crucial for success in algebra and beyond. In the following sections, we'll focus on applying the distributive property to numeric expressions and simplifying them.

Now that we have a solid understanding of the distributive property, let's focus on how to apply it to numeric expressions. Numeric expressions are mathematical phrases that contain numbers and operations, such as addition, subtraction, multiplication, and division. Applying the distributive property to these expressions involves rewriting them in an equivalent form, which can then be simplified to obtain a single numerical value.

The general strategy for applying the distributive property to a numeric expression of the form a(b + c) or a(b - c) is as follows:

  1. Identify the term being distributed: This is the term outside the parentheses, which we'll call 'a'.
  2. Identify the terms inside the parentheses: These are the terms being added or subtracted, which we'll call 'b' and 'c'.
  3. Multiply the term outside the parentheses by each term inside: This means calculating a * b and a * c.
  4. Write the equivalent expression: If the original expression was a(b + c), the equivalent expression is a * b + a * c. If the original expression was a(b - c), the equivalent expression is a * b - a * c.
  5. Simplify the expression: Perform the multiplications and additions or subtractions to obtain a single numerical value.

Let's illustrate this process with an example. Consider the expression 5(3 + 4). To apply the distributive property, we follow the steps outlined above:

  1. The term being distributed is 5.
  2. The terms inside the parentheses are 3 and 4.
  3. Multiply 5 by each term inside: 5 * 3 = 15 and 5 * 4 = 20.
  4. Write the equivalent expression: 5(3 + 4) = 5 * 3 + 5 * 4.
  5. Simplify: 15 + 20 = 35.

Therefore, 5(3 + 4) is equivalent to 35.

It's important to note that the distributive property can also be applied in reverse. This is known as factoring. Factoring involves identifying a common factor in an expression and