Polynomial Division: Finding Quotient & Remainder

Hey there, math enthusiasts! Today, we're diving into the fascinating world of polynomial long division. Specifically, we'll tackle how to divide a polynomial, find its quotient and remainder, and then double-check our work. This is super useful for simplifying expressions, solving equations, and understanding the behavior of functions. Let's get started with our example: dividing $-4x^3 - 3x^2 + 2x - 4$ by $x + 3$.

Understanding Polynomial Long Division

Before we jump into the problem, let's quickly recap what polynomial long division is all about. It's very similar to the long division you learned with numbers, but instead of dividing numbers, we're dividing algebraic expressions. The goal is to find how many times the divisor (the expression we're dividing by) goes into the dividend (the expression being divided). The result gives us a quotient and a remainder. The remainder is what's left over after the division is complete. If the remainder is zero, it means the divisor goes into the dividend perfectly, and we say that the divisor is a factor of the dividend.

Setting Up the Problem

First, we set up our division problem. The dividend, $-4x^3 - 3x^2 + 2x - 4$, goes inside the division symbol, and the divisor, $x + 3$, goes outside. Make sure the terms in the dividend are written in descending order of their exponents (highest power of x first). Luckily, our dividend is already arranged this way. If any terms are missing (like an $x^2$ term), you might want to include them with a coefficient of 0 as a placeholder to keep everything aligned. This helps avoid errors, especially when dealing with multiple variables. For instance, if you were missing the x^2 term, you could write your expression like this: -4x^3 + 0x^2 -3x + 2.

Step-by-Step Long Division

Now, let's walk through the steps of the long division process:

1. Divide the leading terms: Divide the leading term of the dividend ($-4x^3$) by the leading term of the divisor ($x$). $-4x^3 / x = -4x^2$. This is the first term of our quotient. Write this above the $-3x^2$ term in the dividend.
2. Multiply: Multiply the quotient term ($-4x^2$) by the entire divisor ($x + 3$). $-4x^2 * (x + 3) = -4x^3 - 12x^2$.
3. Subtract: Subtract the result from the dividend. This means subtracting $(-4x^3 - 12x^2)$ from $(-4x^3 - 3x^2)$. Remember to change the signs when subtracting. $(-4x^3 - 3x^2) - (-4x^3 - 12x^2) = -4x^3 - 3x^2 + 4x^3 + 12x^2 = 9x^2$. Bring down the next term of the dividend ($+2x$).
4. Repeat: Now, we have a new dividend, $9x^2 + 2x$. Divide the leading term of this new dividend ($9x^2$) by the leading term of the divisor ($x$). $9x^2 / x = 9x$. Write $+9x$ as the next term in the quotient.
5. Multiply: Multiply the new quotient term ($9x$) by the divisor ($x + 3$). $9x * (x + 3) = 9x^2 + 27x$.
6. Subtract: Subtract this result from the current dividend. $(9x^2 + 2x) - (9x^2 + 27x) = 9x^2 + 2x - 9x^2 - 27x = -25x$. Bring down the next term of the dividend ($-4$).
7. Repeat: We now have $-25x - 4$. Divide the leading term of this new dividend ($-25x$) by the leading term of the divisor ($x$). $-25x / x = -25$. Write $-25$ as the next term in the quotient.
8. Multiply: Multiply the new quotient term ($-25$) by the divisor ($x + 3$). $-25 * (x + 3) = -25x - 75$.
9. Subtract: Subtract this result from the current dividend. $(-25x - 4) - (-25x - 75) = -25x - 4 + 25x + 75 = 71$.

We have no more terms to bring down, and the degree of 71 (degree 0) is less than the degree of the divisor (degree 1), so we're done. The quotient is $-4x^2 + 9x - 25$, and the remainder is 71.

Verifying Your Answer

To make sure we've done everything correctly, let's verify our answer. The relationship between the dividend, divisor, quotient, and remainder is expressed by the following equation:

Dividend = Divisor * Quotient + Remainder

Let's plug in our values and see if this holds true:

• Dividend: $-4x^3 - 3x^2 + 2x - 4$
• Divisor: $x + 3$
• Quotient: $-4x^2 + 9x - 25$
• Remainder: $71$

So, our equation becomes:

$-4x^3 - 3x^2 + 2x - 4 = (x + 3)(-4x^2 + 9x - 25) + 71$

Let's expand the right side of the equation:

$(x + 3)(-4x^2 + 9x - 25) = x(-4x^2 + 9x - 25) + 3(-4x^2 + 9x - 25)$

$= -4x^3 + 9x^2 - 25x - 12x^2 + 27x - 75$

Combine like terms:

$= -4x^3 - 3x^2 + 2x - 75$

Now, add the remainder to this result:

$-4x^3 - 3x^2 + 2x - 75 + 71 = -4x^3 - 3x^2 + 2x - 4$

And there we have it! The left side of the equation equals the right side, so our division and remainder are correct. This confirms that we did the polynomial long division accurately.

Conclusion

Polynomial long division might seem a bit tricky at first, but with practice, it becomes much easier. Remember the steps: divide the leading terms, multiply, subtract, and repeat until you can't divide any further. Always double-check your work by multiplying the quotient by the divisor and adding the remainder, and verifying that you get the original dividend. Understanding and mastering polynomial long division opens up a whole new world in algebra, enabling you to solve more complex problems and gain a deeper understanding of mathematical concepts. Keep practicing, and you'll become a pro in no time!

**If you'd like to explore more about polynomial division and related concepts, here's a link to a helpful resource: Khan Academy on Polynomial Division