Problem 26 - Precalculus Mathematics for Calculus

Question

Find the quotient and remainder using synthetic division. $$\frac{x^{2}-5 x+4}{x-1}$$

Step-by-Step Solution

Answer

The quotient is \(x - 4\) with a remainder of 0.

Step 1: Set up the synthetic division

To use synthetic division, write the coefficients of the dividend polynomial, in this case, \(x^2 - 5x + 4\), which are 1, -5, and 4. Next, identify the zero from the divisor \(x-1=0\), which gives \(x=1\). Use this zero in the synthetic division.

Step 2: Begin the synthetic division process

Write down the first coefficient (1) beneath the division line as the starting value for synthetic division. Multiply this by the zero (1) and write the result (1) underneath the next coefficient (-5).

Step 3: Continue the synthetic division process

Add the result from the multiplication (1) to the next coefficient (-5), which gives -4. Multiply this sum (-4) by the zero (1), which gives -4, and write it below the next coefficient (4).

Step 4: Finish the synthetic division process

Add -4 to 4, which results in 0. This value is the remainder of the division. The values written below the line (1 and -4) represent the coefficients of the quotient.

Step 5: Write the quotient and remainder

The quotient is found using the coefficients from the synthetic division results, resulting in \(x - 4\). The remainder of the division process is 0.

Key Concepts

Polynomial Division

Polynomial division is a method similar to long division, used to divide one polynomial by another. It involves finding how many times a smaller polynomial (the divisor) fits into a larger polynomial (the dividend), and detecting any left-over part, known as the remainder.

There are two common techniques for polynomial division:

• Long division: This is just like dividing numbers, matching highest power terms, multiplying, subtracting, and bringing down new terms.
• Synthetic division: A streamlined process used only when dividing by a linear polynomial (of the form \(x-c\)). It's quicker and involves less notation than long division.

Synthetic division is often preferred when applicable because it simplifies the process and reduces errors. It primarily involves manipulating coefficients of the polynomials instead of the variables themselves.

Quotient and Remainder

In polynomial division, an essential goal is to find the quotient and the remainder. The quotient is akin to the answer you get in arithmetic division. It expresses how many times the divisor fits into the dividend without exceeding it.

The remainder is what is left over after the division. In many cases, it may be zero. A zero remainder indicates that the divisor perfectly divides the dividend.

In synthetic division, once the division process is completed, the result is interpreted as follows:

• The numbers at the bottom (except the last term) are the coefficients of the quotient polynomial.
• The last number in the synthetic division table is the remainder.

For instance, in the given example, using synthetic division, the quotient is \(x-4\), and the remainder is 0. This indicates a perfect division.

Divisor and Dividend

To perform a division, we need a dividend and a divisor. In our problem, the dividing expression \(x^2 - 5x + 4\) is the dividend, and the expression \(x-1\) is the divisor.

The dividend is the polynomial you want to divide up. It is the larger polynomial under consideration.

The divisor is the polynomial you are dividing by. Often, in synthetic division, this will be a factor of the form \(x-c\). The real magic of synthetic division happens here because you use the root of the divisor (in our example \(x=1\)) to carry out the division without writing out the full polynomial format.

Understanding these terms is crucial for setting up and executing synthetic division accurately. It ensures that the coefficients are aligned properly and that the division process is applied correctly.