Simplifying And Expanding Algebraic Expressions A Step By Step Guide

This article provides a detailed walkthrough on how to simplify and expand various algebraic expressions. Mastering these skills is crucial for success in algebra and beyond. We will tackle expressions involving squares, products, and combinations of different terms. Let's dive in and learn how to manipulate these expressions effectively.

(a) (a - b)^2 - (a + b)(a - b)

To simplify this expression, we need to expand both the squared term and the product of the binomials. First, let's expand (a - b)^2. This can be done using the formula (x - y)^2 = x^2 - 2xy + y^2. Applying this, we get:

(a - b)^2 = a^2 - 2ab + b^2

Next, we need to expand (a + b)(a - b). This is a difference of squares, which follows the formula (x + y)(x - y) = x^2 - y^2. Applying this, we get:

(a + b)(a - b) = a^2 - b^2

Now, we substitute these expansions back into the original expression:

(a - b)^2 - (a + b)(a - b) = (a^2 - 2ab + b^2) - (a^2 - b^2)

Distribute the negative sign in the second term:

= a^2 - 2ab + b^2 - a^2 + b^2

Combine like terms:

= a^2 - a^2 - 2ab + b^2 + b^2 = -2ab + 2b^2

So, the simplified expression is -2ab + 2b^2. This process involves recognizing patterns, applying algebraic identities, and carefully combining like terms. Understanding these steps is key to simplifying more complex expressions.

(b) (2x - y)^2 + (x + y)(x - y)

In this expression, we again have a squared term and a product of binomials. We'll start by expanding (2x - y)^2. Using the formula (x - y)^2 = x^2 - 2xy + y^2, we substitute 2x for x and y for y:

(2x - y)^2 = (2x)^2 - 2(2x)(y) + y^2 = 4x^2 - 4xy + y^2

Next, we expand (x + y)(x - y), which is another difference of squares:

(x + y)(x - y) = x^2 - y^2

Now, we substitute these expansions back into the original expression:

(2x - y)^2 + (x + y)(x - y) = (4x^2 - 4xy + y^2) + (x^2 - y^2)

Combine like terms:

= 4x^2 + x^2 - 4xy + y^2 - y^2 = 5x^2 - 4xy

Therefore, the simplified expression is 5x^2 - 4xy. This example reinforces the importance of correctly applying the difference of squares formula and the formula for the square of a binomial. Careful attention to the coefficients and variables ensures an accurate simplification.

(c) (a + x)^2 - 2ax + (x - a)(x + a)

This expression involves a squared term, a subtraction, and a product of binomials. Let's start by expanding (a + x)^2. Using the formula (x + y)^2 = x^2 + 2xy + y^2, we get:

(a + x)^2 = a^2 + 2ax + x^2

Next, we expand (x - a)(x + a), which is another difference of squares:

(x - a)(x + a) = x^2 - a^2

Now, we substitute these expansions back into the original expression:

(a + x)^2 - 2ax + (x - a)(x + a) = (a^2 + 2ax + x^2) - 2ax + (x^2 - a^2)

Combine like terms:

= a^2 + 2ax + x^2 - 2ax + x^2 - a^2 = a^2 - a^2 + 2ax - 2ax + x^2 + x^2 = 2x^2

Thus, the simplified expression is 2x^2. This example demonstrates how terms can cancel each other out during the simplification process, leading to a more concise expression. Recognizing and applying the appropriate algebraic identities is crucial.

(d) (a + b)^2 - (a - b)^2 - 4ab

This expression involves two squared terms and a subtraction. We start by expanding (a + b)^2 and (a - b)^2:

(a + b)^2 = a^2 + 2ab + b^2 (a - b)^2 = a^2 - 2ab + b^2

Now, we substitute these expansions back into the original expression:

(a + b)^2 - (a - b)^2 - 4ab = (a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) - 4ab

Distribute the negative sign in the second term:

= a^2 + 2ab + b^2 - a^2 + 2ab - b^2 - 4ab

Combine like terms:

= a^2 - a^2 + 2ab + 2ab - 4ab + b^2 - b^2 = 0

Therefore, the simplified expression is 0. This example highlights how seemingly complex expressions can simplify to a very basic result when terms cancel out. Paying close attention to signs and combining like terms accurately is essential.

(e) (m + n)^2 - 2mn + (n - m)(n + m)

This expression involves a squared term, a subtraction, and a product of binomials. Let's start by expanding (m + n)^2:

(m + n)^2 = m^2 + 2mn + n^2

Next, we expand (n - m)(n + m), which is another difference of squares:

(n - m)(n + m) = n^2 - m^2

Now, we substitute these expansions back into the original expression:

(m + n)^2 - 2mn + (n - m)(n + m) = (m^2 + 2mn + n^2) - 2mn + (n^2 - m^2)

Combine like terms:

= m^2 + 2mn + n^2 - 2mn + n^2 - m^2 = m^2 - m^2 + 2mn - 2mn + n^2 + n^2 = 2n^2

Thus, the simplified expression is 2n^2. This example further illustrates the importance of recognizing patterns and applying algebraic identities to simplify expressions effectively.

(f) (a - b^3)(a + b^3)

This expression is a product of binomials and fits the difference of squares pattern. We can directly apply the formula (x - y)(x + y) = x^2 - y^2. In this case, x = a and y = b^3:

(a - b^3)(a + b^3) = a^2 - (b³⁾2

Simplify the second term:

= a^2 - b^6

Therefore, the simplified expression is a^2 - b^6. This example demonstrates how recognizing the difference of squares pattern can quickly lead to a simplified expression, even when dealing with terms involving exponents.

Simplifying and expanding algebraic expressions is a fundamental skill in mathematics. By understanding and applying algebraic identities, such as the square of a binomial and the difference of squares, we can effectively manipulate and simplify complex expressions. Practice is key to mastering these techniques, and the examples provided in this article offer a solid foundation for further exploration in algebra. Remember to always double-check your work and pay close attention to signs and coefficients. With consistent effort, you can confidently tackle a wide range of algebraic simplification problems.