Polynomial Division Explained Find Quotient And Remainder Of (7x^3-25x^2-17x+17)/(x-4)

Hey guys! Today, we're diving into polynomial division. Specifically, we're going to tackle the problem of dividing the polynomial $7x^3 - 25x^2 - 17x + 17$ by $x - 4$. This might sound intimidating, but don't worry, we'll break it down step by step. Our goal is to find the quotient and the remainder of this division. So, let's get started!

Understanding Polynomial Division

Before we jump into the specific problem, let's quickly recap what polynomial division is all about. Polynomial division is similar to long division with numbers, but instead of digits, we're dealing with terms involving variables and exponents. The main idea is to divide a polynomial (the dividend) by another polynomial (the divisor) to obtain a quotient and a remainder. The relationship between these parts can be expressed as:

$$\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}$$

In our case, the dividend is $7x^3 - 25x^2 - 17x + 17$, and the divisor is $x - 4$. We're on a quest to find the quotient and the remainder. There are a couple of common methods for polynomial division: long division and synthetic division. For this example, we'll use long division because it's a more general method that works for all polynomial divisions, while synthetic division is often quicker but only works when dividing by a linear divisor (like our $x - 4$).

Setting Up the Long Division

Okay, let's set up the long division. It looks similar to the long division you might remember from elementary school, but with polynomials instead of numbers. We write the dividend ($7x^3 - 25x^2 - 17x + 17$) inside the division symbol and the divisor ($x - 4$) outside.

 x - 4 | 7x^3 - 25x^2 - 17x + 17

Now we're ready to roll!

Performing the Long Division Step-by-Step

Here's where the magic happens. We'll walk through the long division process step-by-step. Remember, the key is to focus on the leading terms at each stage.

Step 1: Divide the leading terms

Look at the leading term of the dividend ($7x^3$) and the leading term of the divisor ($x$). What do we need to multiply $x$ by to get $7x^3$? The answer is $7x^2$. So, we write $7x^2$ above the division symbol, in the $x^2$ column.

 7x^2
 x - 4 | 7x^3 - 25x^2 - 17x + 17

Step 2: Multiply the quotient term by the divisor

Now, multiply the $7x^2$ we just wrote down by the entire divisor $(x - 4)$. This gives us $7x^2(x - 4) = 7x^3 - 28x^2$. Write this result below the dividend, aligning like terms.

 7x^2
 x - 4 | 7x^3 - 25x^2 - 17x + 17
 7x^3 - 28x^2

Step 3: Subtract

Subtract the expression we just wrote down ($7x^3 - 28x^2$) from the corresponding terms in the dividend ($7x^3 - 25x^2$). This is crucial – pay attention to the signs! We have $(7x^3 - 25x^2) - (7x^3 - 28x^2) = 7x^3 - 25x^2 - 7x^3 + 28x^2 = 3x^2$. Write the result below.

 7x^2
 x - 4 | 7x^3 - 25x^2 - 17x + 17
 7x^3 - 28x^2
 ---------
 3x^2

Step 4: Bring down the next term

Bring down the next term from the dividend ($-17x$) and write it next to the $3x^2$.

 7x^2
 x - 4 | 7x^3 - 25x^2 - 17x + 17
 7x^3 - 28x^2
 ---------
 3x^2 - 17x

Step 5: Repeat the process

Now we repeat the process with the new expression $3x^2 - 17x$. Divide the leading term $3x^2$ by the leading term of the divisor $x$. We get $3x^2 / x = 3x$. Write $+3x$ next to $7x^2$ in the quotient.

 7x^2 + 3x
 x - 4 | 7x^3 - 25x^2 - 17x + 17
 7x^3 - 28x^2
 ---------
 3x^2 - 17x

Multiply $3x$ by the divisor $(x - 4)$ to get $3x(x - 4) = 3x^2 - 12x$. Write this below $3x^2 - 17x$.

 7x^2 + 3x
 x - 4 | 7x^3 - 25x^2 - 17x + 17
 7x^3 - 28x^2
 ---------
 3x^2 - 17x
 3x^2 - 12x

Subtract $(3x^2 - 17x) - (3x^2 - 12x) = 3x^2 - 17x - 3x^2 + 12x = -5x$.

 7x^2 + 3x
 x - 4 | 7x^3 - 25x^2 - 17x + 17
 7x^3 - 28x^2
 ---------
 3x^2 - 17x
 3x^2 - 12x
 ---------
 -5x

Bring down the next term, $+17$.

 7x^2 + 3x
 x - 4 | 7x^3 - 25x^2 - 17x + 17
 7x^3 - 28x^2
 ---------
 3x^2 - 17x
 3x^2 - 12x
 ---------
 -5x + 17

Repeat the process one more time. Divide the leading term $-5x$ by $x$ to get $-5$. Write $-5$ next to $3x$ in the quotient.

 7x^2 + 3x - 5
 x - 4 | 7x^3 - 25x^2 - 17x + 17
 7x^3 - 28x^2
 ---------
 3x^2 - 17x
 3x^2 - 12x
 ---------
 -5x + 17

Multiply $-5$ by the divisor $(x - 4)$ to get $-5(x - 4) = -5x + 20$. Write this below $-5x + 17$.

 7x^2 + 3x - 5
 x - 4 | 7x^3 - 25x^2 - 17x + 17
 7x^3 - 28x^2
 ---------
 3x^2 - 17x
 3x^2 - 12x
 ---------
 -5x + 17
 -5x + 20

Subtract $(-5x + 17) - (-5x + 20) = -5x + 17 + 5x - 20 = -3$.

 7x^2 + 3x - 5
 x - 4 | 7x^3 - 25x^2 - 17x + 17
 7x^3 - 28x^2
 ---------
 3x^2 - 17x
 3x^2 - 12x
 ---------
 -5x + 17
 -5x + 20
 ---------
 -3

Identifying the Quotient and Remainder

We've reached the end of the long division process! The expression above the division symbol, $7x^2 + 3x - 5$, is the quotient. The value left at the bottom, $-3$, is the remainder. Awesome!

Expressing the Answer

So, we can express the result of the division as:

$$\frac{7x^3 - 25x^2 - 17x + 17}{x - 4} = 7x^2 + 3x - 5 + \frac{-3}{x - 4}$$

Or, we can write it in the form Dividend = (Divisor × Quotient) + Remainder:

$$7x^3 - 25x^2 - 17x + 17 = (x - 4)(7x^2 + 3x - 5) - 3$$

Conclusion

There you have it! We've successfully divided the polynomial $7x^3 - 25x^2 - 17x + 17$ by $x - 4$ using long division. We found the quotient to be $7x^2 + 3x - 5$ and the remainder to be $-3$. Polynomial division might seem tricky at first, but with practice, you'll get the hang of it. Just remember to take it step by step, focus on the leading terms, and pay close attention to the signs.

I hope this explanation was helpful, guys. Keep practicing, and you'll master polynomial division in no time! Good job!