Find The Quotient: (x^4-3x^2+4x-3) / (x^2+x-3)

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the fascinating world of polynomial division. You know, those algebraic expressions with variables and exponents that sometimes feel like a puzzle? Well, we're here to solve one of those puzzles for you. We're going to tackle a problem where we need to find the quotient when a complex polynomial, specifically $\left(x^4-3 x^2+4 x-3\right)$, is divided by a simpler one, $\left(x^2+x-3\right)$. And guess what? The problem statement assures us that the result, the quotient, is indeed a polynomial. This means we're not going to end up with any messy fractions or remainders. Our mission, should we choose to accept it, is to identify this exact polynomial quotient from the given options. So, grab your calculators, your notebooks, and let's get ready to break down this algebraic challenge step-by-step.

Understanding Polynomial Division

Before we jump into solving this specific problem, let's get a solid grasp on what polynomial division actually is. Think of it like regular division, the kind you learned way back when, but with polynomials instead of simple numbers. Just like dividing 10 by 2 gives you 5, dividing one polynomial by another yields a quotient and potentially a remainder. The core idea is to find a polynomial (the quotient) that, when multiplied by the divisor, gets us as close as possible to the dividend without exceeding it, leaving a remainder that has a lower degree than the divisor. This process is fundamental in algebra and has applications in various fields, including calculus, engineering, and computer science. The process itself involves a series of steps that are remarkably similar to long division with numbers. We align the terms by degree, divide the leading terms, multiply the result back into the divisor, subtract, and then bring down the next term. We repeat this until the degree of the remaining polynomial is less than the degree of the divisor. It's a systematic approach that ensures accuracy and helps us understand the relationship between different polynomials. For our specific problem, we're looking for that exact polynomial quotient, so we'll be performing this process with $\left(x^4-3 x^2+4 x-3\right)$ as the dividend and $\left(x^2+x-3\right)$ as the divisor. The assurance that the quotient is a polynomial simplifies things, as we know we'll end up with a clean algebraic expression. Let's get our hands dirty with the actual division.

Step-by-Step Polynomial Long Division

Alright guys, let's get down to business and perform the polynomial long division. We have our dividend: $x^4 - 3x^2 + 4x - 3$ and our divisor: $x^2 + x - 3$. First, it's super important to make sure both polynomials are written in standard form, with terms arranged in descending order of their exponents. If any powers are missing, we should include them with a coefficient of zero. In our case, the dividend is $x^4 + 0x^3 - 3x^2 + 4x - 3$. This step is crucial to avoid errors. Now, let's set up our long division bracket. The divisor, $x^2 + x - 3$, goes on the outside, and the dividend, $x^4 + 0x^3 - 3x^2 + 4x - 3$, goes on the inside.

• Step 1: Divide the leading terms. Take the leading term of the dividend ($x^4$) and divide it by the leading term of the divisor ($x^2$). This gives us $x^4 / x^2 = x^2$. This $x^2$ is the first term of our quotient.

• Step 2: Multiply the quotient term by the divisor. Multiply $x^2$ by the entire divisor $(x^2 + x - 3)$: $x^2(x^2 + x - 3) = x^4 + x^3 - 3x^2$. Write this result below the dividend, aligning terms by their powers.

• Step 3: Subtract. Subtract this product from the dividend. Remember to change the signs of each term in the product when subtracting: $(x^4 + 0x^3 - 3x^2) - (x^4 + x^3 - 3x^2)$. This simplifies to $-x^3 + 0x^2$.

• Step 4: Bring down the next term. Bring down the next term from the dividend ($+4x$) to form the new polynomial: $-x^3 + 0x^2 + 4x$.

Now, we repeat the process with this new polynomial.

• Step 5: Divide the new leading terms. Divide the leading term of the new polynomial ($-x^3$) by the leading term of the divisor ($x^2$): $-x^3 / x^2 = -x$. This $-x$ is the second term of our quotient.

• Step 6: Multiply. Multiply $-x$ by the divisor $(x^2 + x - 3)$: $-x(x^2 + x - 3) = -x^3 - x^2 + 3x$. Write this below the current polynomial.

• Step 7: Subtract. Subtract this product: $(-x^3 + 0x^2 + 4x) - (-x^3 - x^2 + 3x)$. Changing signs gives: $-x^3 + 0x^2 + 4x + x^3 + x^2 - 3x = x^2 + x$.

• Step 8: Bring down the next term. Bring down the last term from the dividend ($-3$): $x^2 + x - 3$.

We repeat the process one more time.

• Step 9: Divide the new leading terms. Divide the leading term of the current polynomial ($x^2$) by the leading term of the divisor ($x^2$): $x^2 / x^2 = 1$. This $1$ is the third term of our quotient.

• Step 10: Multiply. Multiply $1$ by the divisor $(x^2 + x - 3)$: $1(x^2 + x - 3) = x^2 + x - 3$. Write this below the current polynomial.

• Step 11: Subtract. Subtract this product: $(x^2 + x - 3) - (x^2 + x - 3)$. This equals $0$.

Since we have a remainder of $0$, the division is exact, and our quotient is the polynomial we've constructed: $x^2 - x + 1$. That wasn't so bad, right? It's all about following the steps meticulously.

Verifying the Quotient

So, we've performed the polynomial long division and arrived at a quotient of $x^2 - x + 1$. But how do we know for sure that this is the correct answer? The problem statement itself gives us a huge clue: it states that the quotient is a polynomial. This implies that the division should result in a remainder of zero when we multiply our quotient by the divisor. To verify our answer, we can simply multiply our calculated quotient, $(x^2 - x + 1)$, by the original divisor, $(x^2 + x - 3)$. If we get back our original dividend, $\left(x^4-3 x^2+4 x-3\right)$, then we know we've nailed it!

Let's do the multiplication:

$(x^2 - x + 1) \times (x^2 + x - 3)$

We can distribute each term of the first polynomial to the second polynomial:

$x^2(x^2 + x - 3) = x^4 + x^3 - 3x^2$

$-x(x^2 + x - 3) = -x^3 - x^2 + 3x$

$+1(x^2 + x - 3) = x^2 + x - 3$

Now, let's add these results together, combining like terms:

$(x^4 + x^3 - 3x^2) + (-x^3 - x^2 + 3x) + (x^2 + x - 3)$

$= x^4 + (x^3 - x^3) + (-3x^2 - x^2 + x^2) + (3x + x) - 3$

$= x^4 + 0x^3 - 3x^2 + 4x - 3$

$= x^4 - 3x^2 + 4x - 3$

Boom! We got our original dividend back. This confirms that our polynomial quotient is indeed $x^2 - x + 1$. It's always a good practice to double-check your work, especially in math, guys. This verification step ensures accuracy and boosts your confidence in the result. So, the quotient we found is correct!

Analyzing the Options

Now that we've meticulously performed the polynomial division and verified our result, let's take a look at the options provided to see which one matches our answer. Remember, we found our quotient to be $x^2 - x + 1$. Let's examine each option:

• A. $x^4-2 x^2+5 x-6$: This polynomial has a degree of 4. Our calculated quotient has a degree of 2. Since the degree doesn't match, this option is incorrect.

• B. $x^2-x+1$: This polynomial has a degree of 2. The coefficients and terms exactly match our calculated quotient. This is a strong contender!

• C. $x^6+x^5-6 x^4+x^3+10 x^2-15 x+9$: This polynomial has a degree of 6. Our quotient is degree 2. Clearly, this is not our answer.

• D. $x^4-4 x^2+3 x$: This polynomial has a degree of 4. Our quotient is degree 2. This option is also incorrect.

Based on our step-by-step division and verification, the only option that matches our result is B. $x^2-x+1$. This is the correct polynomial quotient. It's always satisfying when your calculated answer aligns perfectly with one of the given choices. It reaffirms that your process was sound and your calculations were accurate. Keep practicing these polynomial divisions, and you'll become a pro in no time!

Conclusion: The Quotient Revealed

So, after carefully navigating the process of polynomial long division, we have successfully determined the quotient of $\left(x^4-3 x^2+4 x-3\right)$ divided by $\left(x^2+x-3\right)$. The journey involved setting up the division correctly, systematically dividing leading terms, multiplying, subtracting, and bringing down subsequent terms until we reached a remainder of zero. This clean remainder confirms that the division is exact, yielding a polynomial quotient. We then took the crucial step of verifying our answer by multiplying the quotient we found, $x^2 - x + 1$, back by the divisor, $x^2 + x - 3$. The multiplication yielded precisely the original dividend, $x^4 - 3x^2 + 4x - 3$, validating our calculation. Comparing our result with the given options, it became clear that option B, $x^2-x+1$, is the correct answer. This problem is a great example of how understanding and applying the principles of polynomial division can lead to clear and definitive solutions. It highlights the importance of attention to detail and systematic execution in algebraic manipulations. Keep practicing these skills, guys, because the more you work with polynomials, the more comfortable and confident you'll become. Remember, every solved problem is a step towards mastering algebraic concepts. Thanks for joining us on Plastik Magazine for this math deep dive!