Simplify Polynomial Combinations

Hey math lovers! Today, we're diving deep into the awesome world of polynomial combinations. You know, those algebraic expressions that can look a little daunting at first glance but are super satisfying to simplify once you get the hang of it. We've got a few challenges lined up for you, and by the end of this, you'll be a polynomial pro. Let's get those brains warmed up and tackle these problems head-on!

Problem 1: Adding Polynomials

Our first challenge involves adding two polynomials: $\left(x^2+3 x-1\right)+\left(x-2 x^2+4\right)$. When you're adding polynomials, the main goal is to combine like terms. Like terms are those terms that have the same variable raised to the same power. Think of them as buddies who always stick together. So, when you see an $x^2$ term, you look for other $x^2$ terms to combine it with. Same goes for $x$ terms and those lonely constant terms (the numbers without any variables). Let's break this one down.

First, let's rearrange the second polynomial so its terms are in descending order of power, just like the first one. This makes it easier to spot the like terms. So, $\left(x-2 x^2+4\right)$ becomes $\left(-2 x^2+x+4\right)$.

Now, we add the two polynomials:

$\qquad (x^2 + 3x - 1) + (-2x^2 + x + 4)$

Let's group the like terms together:

$\qquad (x^2 - 2x^2) + (3x + x) + (-1 + 4)$

Now, perform the addition for each group:

• For the $x^2$ terms: $1x^2 - 2x^2 = -1x^2$ (or just $-x^2$)
• For the $x$ terms: $3x + 1x = 4x$
• For the constant terms: $-1 + 4 = 3$

Putting it all together, the simplified polynomial is: $\mathbf{-x^2 + 4x + 3}$.

Wait a minute! Looking back at the options provided, it seems I might have misread something or perhaps there's a slight variation in the problem presented versus the options. Let me re-check the provided options and the original problem. Ah, I see! It seems the option c. -x^2+4 is presented. Let me re-evaluate the original problem to see if I can match it. The problem is $\left(x^2+3 x-1\right)+\left(x-2 x^2+4\right)$. My calculation led to $-x^2 + 4x + 3$. Option c is $-x^2+4$. These do not match. Let me assume there was a typo in the problem or the options and re-examine the structure. Perhaps the question intended to have an option that exactly matches my result, or perhaps one of the options is intended to be the answer to a slightly different problem. Given the options, let me carefully re-do the addition. $(x^2 + 3x - 1) + (-2x^2 + x + 4)$. Combining $x^2$ terms: $x^2 - 2x^2 = -x^2$. Combining $x$ terms: $3x + x = 4x$. Combining constants: $-1 + 4 = 3$. So, the result is indeed $-x^2 + 4x + 3$. Since none of the options exactly match this, and the instruction is to 'Match each polynomial combination below to the correct simplified polynomial', this suggests a potential issue with the question's provided options. However, if we must choose the closest or assume a typo, option 'c' looks like it might be intended for a problem where the $4x$ term canceled out or was not present. For the sake of proceeding with the exercise as presented, and acknowledging the mismatch, I will note that my derived answer is $-x^2 + 4x + 3$. If I am forced to select from the given options and assuming there might be a typo in the question itself or the options provided, I cannot definitively match it. Let's proceed to the next problem, and I'll re-address this if there's a pattern.

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Problem 2: Subtracting Polynomials

Next up, we have a subtraction problem: $\left(4 x^2-2 x+7\right)-\left(x^2-3 x+4\right)$. Subtracting polynomials is a bit like adding them, but with a crucial extra step: distributing the negative sign. When you subtract a polynomial, you're essentially multiplying each term inside the parentheses by -1. This flips the sign of every term in the polynomial being subtracted. So, that $-(\text{polynomial})$ becomes $+ (\text{opposite of polynomial})$.

Let's write it out:

$\qquad (4x^2 - 2x + 7) - (x^2 - 3x + 4)$

Now, distribute the negative sign to each term inside the second set of parentheses:

$\qquad 4x^2 - 2x + 7 - x^2 - (-3x) - (+4)$

This simplifies to:

$\qquad 4x^2 - 2x + 7 - x^2 + 3x - 4$

Now, we group and combine the like terms, just like we did before:

• $x^2$ terms: $4x^2 - x^2 = 3x^2$
• $x$ terms: $-2x + 3x = 1x$ (or just $x$)
• Constant terms: $7 - 4 = 3$

Putting it all together, the simplified polynomial is: $\mathbf{3x^2 + x + 3}$.

Again, let's check this against the provided options: a. 5x^2+5x-9, b. x^3+2x^2+2x-2, c. -x^2+4. My calculated answer $3x^2 + x + 3$ does not match any of these options. This is quite peculiar! It's possible there's a consistent error in how the problems or options were transcribed. Let me meticulously re-check the subtraction. $\left(4 x^2-2 x+7\right)-\left(x^2-3 x+4\right)$. Change signs: $4x^2 - 2x + 7 - x^2 + 3x - 4$. Combine $x^2$: $4x^2 - x^2 = 3x^2$. Combine $x$: $-2x + 3x = x$. Combine constants: $7 - 4 = 3$. The result $3x^2 + x + 3$ seems correct based on standard algebraic rules. Since none of the options match, I must conclude there's an issue with the question set as presented. I will proceed to the third problem and note this discrepancy.

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Problem 3: Multiplying and Adding Polynomials

Our final challenge involves a mix of multiplication and addition: $5(x-1)(x+2)+1$. This one requires us to first multiply the two binomials $(x-1)$ and $(x+2)$, then multiply that result by 5, and finally add 1. We'll use the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last) for multiplying two binomials.

Let's start by multiplying $(x-1)(x+2)$:

• First: $x imes x = x^2$
• Outer: $x imes 2 = 2x$
• Inner: $-1 imes x = -x$
• Last: $-1 imes 2 = -2$

Combine these terms: $x^2 + 2x - x - 2$.

Now, combine the like terms ($2x$ and $-x$):

$\qquad x^2 + (2x - x) - 2 = x^2 + x - 2$

Great! Now we need to multiply this result by 5:

$\qquad 5(x^2 + x - 2)$

Distribute the 5 to each term inside the parentheses:

$\qquad 5 imes x^2 + 5 imes x + 5 imes (-2)$

$\qquad 5x^2 + 5x - 10$

Finally, we add 1:

$\qquad (5x^2 + 5x - 10) + 1$

Combine the constant terms ($-10$ and $+1$):

$\qquad 5x^2 + 5x + (-10 + 1)$

$\qquad \mathbf{5x^2 + 5x - 9}$

Alright, let's see if this one matches an option! Looking at the choices: a. 5x^2+5x-9, b. x^3+2x^2+2x-2, c. -x^2+4. Yes! Our calculated result, $5x^2 + 5x - 9$, perfectly matches option a. Hooray!

Putting It All Together (and Addressing the Discrepancies)

So, we successfully solved the third problem and matched it to option a. However, for the first two problems, my calculations led to results that were not present in the given options. Let me list my derived results again:

1. For \left(x^2+3 x-1 ight)+\left(x-2 x^2+4\right), I calculated $\mathbf{-x^2 + 4x + 3}$. Option c was $-x^2+4$. It seems very close, differing only by the $+4x$ term. It's plausible that either the original problem statement had a typo (e.g., the $3x$ was meant to be $x$, or the $x$ in the second polynomial was meant to be $-3x$), or option c is a simplified version that dropped a term by mistake.
2. For \left(4 x^2-2 x+7 ight)-\left(x^2-3 x+4\right), I calculated $\mathbf{3x^2 + x + 3}$. None of the options were close to this result. Option a was $5x^2+5x-9$, b was $x^3+2x^2+2x-2$, and c was $-x^2+4$. It's possible there was a significant transcription error in the question or options for this problem.

It's super important in math, guys, to be meticulous. Double-checking your work is key, especially when dealing with signs and multiple steps. If you ever find yourself in a situation where your answer doesn't match the options, don't panic! Take a deep breath, re-read the question, re-do your calculations step-by-step, and consider if there might be a typo in the provided materials. Sometimes, the error isn't yours; it's in the question itself!

For this specific exercise, given the discrepancy, the most accurate way to present the matches would be:

• \left(x^2+3 x-1 ight)+\left(x-2 x^2+4 ight) --> My result: $-x^2 + 4x + 3$ (Closest option: c. -x^2+4, assuming a typo)
• \left(4 x^2-2 x+7 ight)-\left(x^2-3 x+4 ight) --> My result: $3x^2 + x + 3$ (No matching option found)
• $5(x-1)(x+2)+1$ --> Matches option a. 5x^2+5x-9

Keep practicing, and don't let these little hiccups discourage you. Every problem solved, even the ones with errors, helps build your math muscle! Happy calculating!