Simplifying Algebraic Expression A(x + A)³ − A(x − A)² + X(a − X)³

July 27, 2025 by Scholario Team 67 views

Simplifying the Algebraic Expression a(x + a)³ − a(x − a)² + x(a − x)³ Step-by-Step

Hey guys! Today, we're going to dive into simplifying a pretty hefty algebraic expression. Don't worry, we'll take it step-by-step so it's super easy to follow. The expression we're tackling is: a(x + a)³ − a(x − a)² + x(a − x)³. Sounds intimidating, right? But trust me, with a bit of algebraic maneuvering, we can tame this beast. So, grab your pencils, and let's get started!

Breaking Down the Expression

Before we jump into the nitty-gritty, let's first understand what we're dealing with. This expression involves variables (a and x), exponents, and multiple terms. Our main goal here is to expand the terms, simplify, and combine like terms to get the expression into its simplest form. Remember, the order of operations (PEMDAS/BODMAS) is our best friend here: Parentheses/Brackets, Exponents/Orders, Multiplication and Division, and Addition and Subtraction.

Let's take it piece by piece:

Expanding (x + a)³: This is a binomial cubed, and we'll need to use either the binomial theorem or good ol' multiplication to expand it. We'll go through this in detail shortly.
Expanding (x − a)²: This is a binomial squared, which is a bit simpler. We'll use the formula (A - B)² = A² - 2AB + B².
Expanding (a − x)³: Similar to the first term, this is a binomial cubed, but with a subtraction. We'll need to be careful with the signs here.

Once we've expanded these, we'll multiply them by their respective coefficients (a or x) and then combine all the terms. Sounds like a plan? Awesome, let's dive into the first part!

Expanding (x + a)³

Okay, so let's start with the first challenging part: expanding (x + a)³. This means we need to multiply (x + a) by itself three times: (x + a)(x + a)(x + a). We can approach this in a couple of ways. One way is to first multiply (x + a)(x + a), and then multiply the result by (x + a) again. Another way, and perhaps a more efficient way for those familiar, is to use the binomial theorem or Pascal's Triangle. Let's go through both methods so everyone's on the same page.

Method 1: Step-by-Step Multiplication

First, let's multiply (x + a)(x + a). This is a straightforward application of the distributive property (or the FOIL method, if you prefer):

(x + a)(x + a) = x(x + a) + a(x + a) = x² + ax + ax + a² = x² + 2ax + a²

Now we have x² + 2ax + a², and we need to multiply this by (x + a):

(x² + 2ax + a²)(x + a) = x²(x + a) + 2ax(x + a) + a²(x + a)

Let's distribute again:

= x³ + x²a + 2ax² + 2a²x + a²x + a³

Now, we combine the like terms:

= x³ + 3ax² + 3a²x + a³

So, (x + a)³ = x³ + 3ax² + 3a²x + a³. Whew! That was a bit of work, but we got there. This expansion is crucial, so make sure you're comfortable with it.

Method 2: Using the Binomial Theorem

The binomial theorem provides a formula for expanding binomials raised to a power. For (x + a)³, the binomial theorem tells us:

(x + a)³ = (1)x³a⁰ + (3)x²a¹ + (3)xa² + (1)x⁰a³

The coefficients (1, 3, 3, 1) come from Pascal's Triangle or the binomial coefficients formula. Simplifying this, we get:

(x + a)³ = x³ + 3x²a + 3xa² + a³

Which is the same result we got from step-by-step multiplication! Choose whichever method clicks best for you.

Expanding (x − a)²

Great, we've conquered the cube! Now, let's move on to the next term: (x − a)². This one is a bit more straightforward. Remember the formula for squaring a binomial: (A - B)² = A² - 2AB + B². In our case, A = x and B = a. So, let's plug it in:

(x − a)² = x² - 2(x)(a) + a² = x² - 2ax + a²

See? Much simpler! This expansion is a key component in simplifying our overall expression. Make sure to keep track of this result.

Expanding (a − x)³

Alright, let's tackle the last binomial expansion: (a − x)³. This is similar to (x + a)³ but with a subtraction, so we need to be extra careful with the signs. We can use either the step-by-step multiplication method or the binomial theorem, just like before. For the sake of variety, let's use the binomial theorem again.

The binomial theorem gives us:

(a − x)³ = (1)a³(-x)⁰ + (3)a²(-x)¹ + (3)a¹(-x)² + (1)a⁰(-x)³

Simplifying this, we get:

= a³ - 3a²x + 3ax² - x³

Notice how the signs alternate because of the -x term. This is a critical detail to keep in mind. So, (a − x)³ = a³ - 3a²x + 3ax² - x³. Awesome! We've expanded all the binomials. Now comes the fun part: putting it all together.

Putting It All Together: The Big Kahuna

Okay, guys, we've done the hard work of expanding each term. Now it's time to plug everything back into the original expression and simplify. Remember the original expression:

a(x + a)³ − a(x − a)² + x(a − x)³

Let's substitute our expanded forms:

a(x³ + 3ax² + 3a²x + a³) − a(x² - 2ax + a²) + x(a³ - 3a²x + 3ax² - x³)

Now, we need to distribute the a and x:

= ax³ + 3a²x² + 3a³x + a⁴ − ax² + 2a²x − a³ + a³x - 3a²x² + 3ax³ - x⁴

Looks like a jumbled mess, right? But don't worry, we'll sort it out by combining like terms. This is where the magic happens! Let's group terms with the same variables and exponents together.

Combining Like Terms: Taming the Beast

Time to bring order to the chaos! Let's identify and combine like terms. This involves looking for terms with the same variables raised to the same powers. Here's our expanded expression again:

ax³ + 3a²x² + 3a³x + a⁴ − ax² + 2a²x − a³ + a³x - 3a²x² + 3ax³ - x⁴

Let's rearrange and group the like terms:

= (-x⁴) + (ax³ + 3ax³) + (3a²x² - 3a²x²) + (- ax²) + (3a³x + 2a²x + a³x) + (a⁴ - a³)

Now, let's combine the coefficients:

= -x⁴ + 4ax³ - ax² + (3a³x + 2a²x + a³x) + a⁴ - a³

Oops! I noticed a small error. 3a³x should be 4a³x. I missed adding 1a³x, let's fix that:

= -x⁴ + 4ax³ - ax² + (4a³x + 2a²x) + a⁴ - a³

Let's rearrange the equation for a cleaner look:

= -x⁴ + 4ax³ - ax² + 4a³x + 2a²x + a⁴ - a³

And there we have it! We've successfully combined all the like terms. This is the simplified form of our original expression.

The Final Simplified Expression

After all that hard work, our simplified expression is:

-x⁴ + 4ax³ - ax² + 4a³x + 2a²x + a⁴ - a³

Guys, that was quite a journey! We took a complex algebraic expression and, step-by-step, broke it down, expanded it, and simplified it. Remember, the key to simplifying algebraic expressions is to take it one step at a time, be careful with your signs, and don't be afraid to double-check your work.

Key Takeaways

Binomial Expansion: Mastering binomial expansions, like (x + a)³ and (x − a)², is crucial for simplifying many algebraic expressions.
Combining Like Terms: This is the bread and butter of simplification. Make sure you're grouping terms with the same variables and exponents.
Order of Operations: Always follow PEMDAS/BODMAS to ensure you're simplifying in the correct order.
Double-Check Your Work: Algebraic simplification can be tricky, so take the time to review your steps and catch any errors.

I hope this step-by-step guide has been helpful! Keep practicing, and you'll become an algebraic simplification master in no time. If you have any questions, feel free to ask. Happy simplifying!