Problem 31 - Algebra and Trigonometry

Question

25-38 . Find the quotient and remainder using synthetic division. $$ \frac{x^{3}-8 x+2}{x+3} $$

Step-by-Step Solution

Answer

Quotient: \(x^2 - 3x + 1\), Remainder: \(-1\).

Step 1: Set Up the Synthetic Division

To perform synthetic division, write the coefficients of the polynomial being divided. This polynomial is \(x^3 - 8x + 2\). The coefficients are 1, 0 (as there is no \(x^2\) term), -8, and 2. Write them in order: 1, 0, -8, 2. The divisor is \(x + 3\), so the root used in synthetic division is \(-3\).

Step 2: Perform Synthetic Division Calculations

Write the \(-3\) to the left of a vertical bar. Draw a horizontal line, and write the coefficients 1, 0, -8, 2 to the right of the vertical bar.Bring down the first coefficient (1) to below the line.Now, multiply this number (1) by \(-3\) and write it under the next coefficient (0), then add: 0 + (-3) = -3. Repeat this process:- Multiply -3 by -3, which gives 9. Write 9 under -8, and add: -8 + 9 = 1.- Multiply 1 by -3, which gives -3. Write -3 under 2, and add: 2 + (-3) = -1.

Step 3: Interpret the Results

The numbers under the line, from left to right, are the coefficients of the quotient of the division, and the last number is the remainder.This means the quotient is \(1x^2 - 3x + 1\), and the remainder is \(-1\).

Step 4: Write the Final Answer

The result of the synthetic division is given by the quotient and remainder. The division produces:\[ \frac{x^3 - 8x + 2}{x + 3} = x^2 - 3x + 1 - \frac{1}{x+3} \]

Key Concepts

Polynomial Division

Polynomial division is a method used to divide polynomials, much like how we divide numbers. It helps in determining how a polynomial (the dividend) can be expressed in terms of another polynomial (the divisor) and possibly a remainder.
In this case, we're working with the dividend polynomial \(x^3 - 8x + 2\) and the divisor \(x + 3\).

There are different methods for polynomial division, such as long division and synthetic division. These methods are used depending on the specific problem and the form of the divisor.

• Long division works in a similar way to numeric long division.
• Synthetic division is a shortcut method suited for divisors of the form \(x - c\), though it can be adapted for \(x + c\) by using the opposite value of \(c\).

Synthetic division requires the divisor to be in the form \(x - r\). For \(x + 3\), we use \(-3\) as the root in synthetic division.
This method is often quicker and easier than long division, especially when dealing with higher degree polynomials or when the divisor is a binomial of degree one.

Quotient and Remainder

In any division problem, whether with numbers or polynomials, the result typically consists of a quotient and possibly a remainder. This applies to polynomial division as well.
The quotient is the result of the division that indicates how many times the divisor can "fit" into the dividend. In this exercise, through synthetic division, we find the quotient to be \(x^2 - 3x + 1\).

The remainder, on the other hand, is what's "left over" after the division has been carried out. Synthesizing the calculations gives us a remainder of \(-1\). This means when \(x^3 - 8x + 2\) is divided by \(x + 3\), there's an extra bit of \(-1\) that doesn't complete another whole division.

• The quotient can often be a polynomial of lower degree than the original dividend.
• The remainder is a constant if the divisor is linear, as in this scenario.

In general algebraic expression, the division can be represented as:\[ \frac{Dividend}{Divisor} = Quotient + \frac{Remainder}{Divisor} \]For the given problem, it translates to:\[ \frac{x^3 - 8x + 2}{x + 3} = x^2 - 3x + 1 - \frac{1}{x+3} \]This format makes it easy to understand the distribution of the polynomial elements in the equation.

Algebraic Expressions

Algebraic expressions form the building blocks of algebra. They consist of variables, coefficients, and arithmetic operations, often combined to represent complex mathematical situations.
In polynomial division, understanding the structure of algebraic expressions helps in simplifying and manipulating them efficiently. The expression we started with is \(x^3 - 8x + 2\), where each term is a component of the polynomial that aligns with its power degree.

Here's a breakdown:

• The term \(x^3\) indicates a cubic polynomial, being the highest degree in this expression.
• The term \(-8x\) has no power other than one, representing a linear component, and the lack of an \(x^2\) term signals a zero coefficient for that degree.
• The constant term \(+2\) is independent and adds a fixed value to the expression.

Recognizing how each part of an algebraic expression contributes to the overall polynomial aids in division. It clarifies how coefficients are used in operations like synthetic division and supports performing operations that handle algebraic expressions efficiently and accurately.
Ultimately, learning to manage algebraic expressions through division opens pathways to solving more complex algebra problems.