Solving Complex Algebraic Expressions: A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of algebraic expressions. We're going to tackle some pretty complex calculations step by step. So, grab your pencils and let's get started!

a) 2(3x²+x-2)+(-24x): (-2x²)² - 2(x-5)

Let's break down this expression piece by piece. This is super important, guys, because algebra is like building with LEGOs – you gotta get the foundation right!

First things first, we'll distribute the constants outside the parentheses:

2(3x²) becomes 6x² 2(x) becomes 2x 2(-2) becomes -4 -2(x) becomes -2x -2(-5) becomes +10 So, the expression now looks like this: 6x² + 2x - 4 + (-24x): (-2x²)² - 2x + 10. It's like untangling a ball of yarn, you know? Slow and steady.

Next up, we deal with the exponent and division. Remember your order of operations (PEMDAS/BODMAS)?

(-2x²)² means (-2x²) multiplied by itself, which equals 4x⁴. It's like squaring a number, but with variables. Cool, right? Now, we've got (-24x) : (4x⁴). This is where things get interesting. When dividing terms with exponents, we subtract the exponents. So, (-24x) divided by (4x⁴) is -6/x³. This is because -24 divided by 4 is -6, and x divided by x⁴ is 1/x³, which we can write as x⁻³ or -6/x³.

Our expression now looks like this: 6x² + 2x - 4 - 6/x³ - 2x + 10. We're getting there, guys! Almost there!

Now, let’s simplify by combining like terms. This is like sorting your socks after laundry – putting the pairs together.

We have +2x and -2x, which cancel each other out. Poof! Gone! We also have -4 and +10, which combine to give us +6. So, our expression simplifies to: 6x² - 6/x³ + 6. This is our final simplified expression for part (a). High five!

In summary, for part a), we meticulously applied the distributive property, tackled exponents and division, and then combined like terms to arrive at the simplified form: 6x² - 6/x³ + 6. Remember, each step is a piece of the puzzle. Don't rush, and double-check your work!

b) 3(2x²-x+4)+3(x-5)+(-96x): (-4x²)²

Alright, let's jump into the next expression! It looks a bit similar, but each one has its own little quirks. It's like each math problem is a unique fingerprint.

First, just like before, we distribute the constants across the parentheses:

3(2x²) becomes 6x² 3(-x) becomes -3x 3(4) becomes 12 3(x) becomes 3x 3(-5) becomes -15 So far, so good! Our expression is now: 6x² - 3x + 12 + 3x - 15 + (-96x): (-4x²)². We're on a roll!

Next up is the exponent and division. Remember our friend PEMDAS/BODMAS?

(-4x²)² means (-4x²) multiplied by itself, which equals 16x⁴. Just like before, we're squaring a term with a variable. Now, we have (-96x) : (16x⁴). Time for some division! -96 divided by 16 is -6, and x divided by x⁴ is 1/x³, which is x⁻³. So, (-96x) / (16x⁴) = -6/x³. Our expression now looks like: 6x² - 3x + 12 + 3x - 15 - 6/x³. We're making progress, guys!

Time to combine those like terms! This is where we tidy things up and make the expression look its best.

We have -3x and +3x, which, you guessed it, cancel each other out. Poof! They’re gone! We also have +12 and -15, which combine to give us -3. So, our simplified expression is: 6x² - 3 - 6/x³. There we have it! Another one bites the dust!

To recap, in this part, we again employed the distributive property, simplified exponents and division, and then combined similar terms to obtain the concise form: 6x² - 3 - 6/x³. Keep practicing, and these will become second nature!

c) 4[2x³-3x(x-1)] - 2x[x² + 1-3(x − 1)]

Okay, this one looks a bit more intricate with the nested parentheses and brackets. But don’t worry, we’ve got this! We’ll just take it one step at a time. Think of it like a puzzle with more pieces, but still solvable!

First, let's tackle the inner parentheses. We'll distribute within those first:

-3x(x - 1) becomes -3x² + 3x. Remember to distribute that negative sign! -3(x - 1) becomes -3x + 3. Again, careful with the negative! Now, our expression looks like: 4[2x³ - 3x² + 3x] - 2x[x² + 1 - 3x + 3]. We've cleared the first hurdle!

Next, we simplify within the brackets:

In the second bracket, we combine like terms: 1 + 3 becomes 4. So, our expression is now: 4[2x³ - 3x² + 3x] - 2x[x² - 3x + 4]. Looking good, guys!

Now, we distribute the constants and terms outside the brackets:

4[2x³ - 3x² + 3x] becomes 8x³ - 12x² + 12x -2x[x² - 3x + 4] becomes -2x³ + 6x² - 8x. Again, pay close attention to those signs! Our expression is now: 8x³ - 12x² + 12x - 2x³ + 6x² - 8x. We're in the home stretch!

Finally, we combine like terms. The grand finale!

We have 8x³ and -2x³, which combine to give us 6x³. We have -12x² and +6x², which combine to give us -6x². We have +12x and -8x, which combine to give us +4x. So, our fully simplified expression is: 6x³ - 6x² + 4x. Nailed it!

In this more complex part, the critical steps involved addressing inner parentheses first, simplifying within brackets, carefully distributing terms, and then combining similar terms to arrive at the simplified expression: 6x³ - 6x² + 4x. Patience and precision are your best friends here.

d) 2[3x²(x-1)-x(x-5)] - 3x[2x(x + 1) + 3]

Last but not least, let's tackle the final expression! We’ve come so far, let’s finish strong! This one has a similar structure to the previous one, so we can use the same strategies. Think of it as the final boss level – you’ve got all the skills you need!

First, we distribute within the innermost parentheses:

3x²(x - 1) becomes 3x³ - 3x² -x(x - 5) becomes -x² + 5x. Don’t forget the negative sign distribution! 2x(x + 1) becomes 2x² + 2x Our expression now looks like: 2[3x³ - 3x² - x² + 5x] - 3x[2x² + 2x + 3]. We’ve cleared the first layer!

Next, we simplify within the brackets by combining like terms:

In the first bracket, -3x² and -x² combine to give us -4x². Our expression is now: 2[3x³ - 4x² + 5x] - 3x[2x² + 2x + 3]. Looking good!

Now, we distribute the terms outside the brackets:

2[3x³ - 4x² + 5x] becomes 6x³ - 8x² + 10x -3x[2x² + 2x + 3] becomes -6x³ - 6x² - 9x. Careful with those signs! Our expression is now: 6x³ - 8x² + 10x - 6x³ - 6x² - 9x. Almost there!

Finally, we combine like terms. The final simplification!

We have 6x³ and -6x³, which cancel each other out. Poof! Gone! We have -8x² and -6x², which combine to give us -14x². We have +10x and -9x, which combine to give us +x. So, our final simplified expression is: -14x² + x. Victory is ours!

In this final part, we reinforced our skills by carefully handling nested parentheses, distributing terms, combining similar terms, and arriving at the simplified form: -14x² + x. Congratulations, you’ve conquered the algebraic expressions!

By breaking down each expression step by step, using the distributive property, simplifying exponents and division, and combining like terms, we were able to solve these complex algebraic expressions. Remember, practice makes perfect! Keep at it, and you'll become an algebra whiz in no time! You got this, guys!