Problem 24 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Quotient: \(x - 4\); Remainder: 0.

Step 1: Write Down the Polynomial Coefficients

To start synthetic division, identify the coefficients of the polynomial dividend. For the polynomial \(x^2 - 5x + 4\), the coefficients are 1 for \(x^2\), -5 for \(x\), and 4 for the constant term. This gives us the list: [1, -5, 4].

Step 2: Identify the Divisor

The divisor is \(x - 1\), which gives us \(x - k\) for synthetic division. Set \(x - 1 = 0\) to find \(k\). Here, \(k = 1\).

Step 3: Set Up the Synthetic Division

Write down the coefficients from Step 1. Draw a vertical line to their right and then below the first coefficient, write 0, which is the initial part of the synthetic division process. Place the value of \(k\) (which is 1, from Step 2) outside to the left of the vertical line.

Step 4: Begin Division: Bring Down the First Coefficient

Bring down the first coefficient (1) directly below itself into the bottom row. This number becomes the leading coefficient of the quotient.

Step 5: Multiply and Add

Multiply the number just written (1) by \(k\) (which is 1), and write the result under the next coefficient (-5). Accordingly, 1 times 1 equals 1. Then add this result (-5 + 1) to get -4. Write -4 beneath the line.

Step 6: Repeat Multiply and Add

Repeat the multiply and add process. Multiply -4 by \(k\) (1), resulting in -4. Add this to the next coefficient (4), resulting in 0, which goes in the bottom row.

Step 7: Determine Quotient and Remainder

The bottom row numbers (besides the last one) represent the coefficients of the quotient polynomial. Here, they are 1 and -4, so the quotient is \(x - 4\). The last number in the row, 0, is the remainder.

Key Concepts

Polynomial Division

Polynomial division is a method designed to divide one polynomial by another, similar to the way we divide numbers. In most cases, you will deal with a simpler division process called synthetic division when the divisor is in the form of \(x - k\). In polynomial division, you identify the dividend — the polynomial you are dividing into — and the divisor — the polynomial you are dividing by.

To perform polynomial division:

• Write the dividend (the polynomial you want to divide) and the divisor (what you want to divide by).
• If the divisor is a simple binomial like \(x - k\), synthetic division provides a quick and efficient way to find the quotient and remainder.
• Synthetic division is commonly used if the divisor is linear because it simplifies the arithmetic involved.

By the time you finish the division process, you've essentially subtracted the effect of the divisor enough times that you're left with a quotient and sometimes a remainder.

Quotient and Remainder

In the process of division, both in simple arithmetic and with polynomials, two main results are produced—the quotient and the remainder. Understanding these is essential to mastering division of algebraic expressions.

Quotient
The quotient represents the main result of the division. In polynomial division, after completing your calculations, the numbers arranged in the lower row (excluding the last one) represent the coefficients of the quotient polynomial. For example, in the problem provided, the coefficients [1, -4] translate into the quotient polynomial \(x - 4\).

Remainder
The remainder is what is left over after dividing. In the case of polynomial division, it is the last number left in your solution. Here, it was 0, meaning the division was exact with no remainder. If the remainder had been, say 5, the division would be expressed with the remainder, such as \(x - 4 + \frac{5}{x-1}\).

The idea is simple: the divisor goes into the dividend the number of times stated in the quotient, plus the remainder.

Algebraic Expressions

Algebraic expressions form the basis of polynomial division and they consist of variables, coefficients, and constants combined through addition, subtraction, multiplication, division, or powers. Recognizing the parts of an expression is a crucial step in algebra.

Consider

• The term "monomial" refers to a single algebraic expression, like \(2x\).
• A "binomial" consists of two terms, like \(x + 2\).
• A "trinomial" contains three terms, such as \(x^2 - 5x + 4\).

In the exercise, \(x^2 - 5x + 4\) is a trinomial. Each component of this polynomial plays a role when you perform operations like division.

Understanding algebraic expressions becomes especially crucial when dividing them since you must identify each part accurately to work with them in processes like synthetic division. This knowledge lays the foundation for recognizing patterns, simplifying expressions, and efficiently solving equations.