Problem 31 - Intermediate Algebra

Question

Perform each division. \(\frac{16 x^{2}-16 x-5}{4 x+1}\)

Step-by-Step Solution

Answer

The result of the division is \(4x - 5\).

Step 1: Set Up the Division

To divide the polynomial \(16x^2 - 16x - 5\) by \(4x + 1\), we start by arranging the terms in standard form. The dividend is \(16x^2 - 16x - 5\) and the divisor is \(4x + 1\). Write this as a long division format.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend by the leading term of the divisor. Divide \(16x^2\) by \(4x\) to get \(4x\). This will be the first term of the quotient.

Step 3: Multiply and Subtract

Multiply \(4x\) (the quotient from the previous step) by the entire divisor \(4x + 1\) to get \(16x^2 + 4x\). Subtract this result from the dividend to get a new dividend: \(-20x - 5\).

Step 4: Repeat the Process

Now divide the new leading term \(-20x\) by the leading term of the divisor \(4x\), which gives \(-5\). Add \(-5\) as the next term of the quotient.

Step 5: Multiply and Subtract Again

Multiply \(-5\) by \(4x + 1\) to get \(-20x - 5\). Subtract this from the new dividend \(-20x - 5\) to get a remainder of \(0\).

Step 6: Conclude the Division

Since the remainder is zero, the division is complete. The quotient is \(4x - 5\).

Key Concepts

Long Division

The long division method is not just for numbers; it plays a vital role in polynomial division as well. Polynomial long division helps us break down complex polynomials by dividing them into simpler parts. The method works quite similarly to the division of numbers.

• Write down the dividend, which is the polynomial you want to divide.
• Write the divisor outside the division bar.
• Use a strategy of dividing, multiplying, and subtracting iteratively.

Breaking it into these manageable steps allows us to systematically reduce the polynomial until it cannot be divided any further. Long division of polynomials is essential, especially when simplifying rational expressions or finding asymptotes of functions.

Quotient

The quotient in polynomial division is much like the quotient in basic arithmetic division. It's what you get when you divide the dividend by the divisor. In simpler terms, it is the result of the division process.
When performing polynomial long division:

• Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient.
• Multiply the entire divisor by this first term of the quotient.
• Subtract this product from the original dividend. The result is the new dividend.
• Repeat these steps until the degree of the new dividend is less than the degree of the divisor.

For our example, the quotient we determined was: \(4x - 5\). This represents the simplified version of the original polynomial after division.

Remainder

In polynomial division, the remainder is what is left after the division process is complete. It is similar to the leftover in numerical division when the dividend isn't perfectly divisible by the divisor.

• If the remaining polynomial's degree is less than the divisor's degree, this becomes the remainder.
• A remainder of zero indicates that the dividend is exactly divisible by the divisor.

After performing long division, if you end up with a non-zero remainder, it can often be expressed as a fraction where the remainder is the numerator and the original divisor is the denominator. But in our exercise, we ended with a remainder of 0, indicating a perfect division.

Leading Term

The leading term in a polynomial is the one with the highest degree; it sets the standard for what follows in the long division process. It guides how each division step will proceed during polynomial long division.
Here’s how the leading term is used:

• The leading term of the dividend is divided by the leading term of the divisor to find the first term of the quotient.
• This division is critical as it defines the leading term for the quotient, setting the path for the rest of the division steps.

For instance, in the original division problem, the leading term \(16x^2\) was divided by \(4x\), resulting in the first term of our quotient \(4x\). Using the leading term ensures accuracy and efficiency in simplifying polynomials.