Problem 2 - College Algebra and Calculus: An Applied Approach

Question

Use long division to divide. Divisor \(x-4\) Dividend $$5 x^{2}-17 x-12$$

Step-by-Step Solution

Answer

The quotient when \(5x^2 - 17x - 12\) is divided by \(x - 4\) is \(5x - 3\).

Step 1: Setup of long division

Set up the long division as follows: Write the dividend \(5x^2 - 17x - 12\) inside a long division symbol and the divisor \(x - 4\) outside. The goal is to subtract the dividend from products of the divisor until what’s left (the remainder) is less than the divisor.

Step 2: Divide

First, divide the leading term of the dividend by the leading term of the divisor, i.e., \(5x^2\) divided by \(x\) gives \(5x\). This is the first term of the quotient.

Step 3: Multiply and Subtract

Now, multiply the divisor \(x - 4\) by the first term of quotient \(5x\) we got in step 2, which gives \(5x^2 - 20x\) and subtract this from the dividend. Subtracting gives \(-3x - 12\).

Step 4: Repeat

Repeat the process now with our new dividend \(-3x - 12\). Divide the leading term of the new dividend by the leading term of the divisor, that gives \(-3\). So, the next term of our quotient is \(-3\). Multiply the divisor \(x - 4\) by \(-3\) and subtract it from \(-3x - 12\), which gives 0. So, our division process ends here.

Step 5: Write Final Answer

Write down the final quotient. The final quotient is the expression obtained above which is \(5x - 3\). And the remainder is 0.

Key Concepts

Quotient

In polynomial long division, the quotient is the result you get when you divide two polynomials. Think of it like the result you get when you divide two numbers. In this specific exercise, we are dividing the polynomial \(5x^2 - 17x - 12\) by \(x - 4\). As you perform the division, you subtract out multiples of the divisor from the dividend, and the coefficients you use make up the terms of the quotient.

Here's a quick breakdown of how we arrive at the quotient:

• First, divide the leading term of the dividend, \(5x^2\), by the leading term of the divisor, \(x\), which gives \(5x\). This becomes the first term in the quotient.
• Then, for the next step, repeat the division using the new polynomial formed by the subtraction. This smaller polynomial is used to determine the next term in the quotient.
• In this case, the next term is \(-3\).
• A complete division results in the final quotient \(5x - 3\).

The quotient reflects the number and type of times the divisor can "fit" into the dividend fully.

Divisor

The divisor is a key part of polynomial division, as it is the polynomial you are dividing by. It operates just like the number you use to divide in ordinary arithmetic long division. In our given problem, the divisor is \(x - 4\).

• The process begins with comparing the leading term of the divisor and the dividend.
• We use the divisor to determine how many times it "fits" into the leading part of the dividend.
• This helps us figure out the individual terms of the quotient.
• As we continue dividing into new sub-dividends, the role of divisor remains consistent and pivotal in narrowing down what each new term of the quotient becomes.

A strong understanding of how to apply the divisor repeatedly is crucial for successfully completing the division.

Remainder

During polynomial division, the remainder is what is left over after dividing the dividend completely by the divisor. After the series of subtractions in the division are complete, the remainder is the last piece left over that is less than the degree of the divisor.

• In this example, since \(5x^2 - 17x - 12\) divides perfectly by \(x - 4\), the remainder turns out to be 0.
• If the remainder is non-zero, it will be of a lower degree than the divisor.
• In some problems, having a non-zero remainder means we would express the final result as the quotient plus the remainder over the divisor.

A zero remainder indicates the dividend is exactly divisible by the divisor, which is the case in our problem, confirming that \(5x - 3\) is the complete and final quotient.