Problem 46 - College Algebra with Modeling and Visualization

Question

Use synthetic division to divide the first polymomial by the second. $$x^{4}-\frac{1}{2} x^{3}+3 x^{2}-\frac{5}{2} x+\frac{9}{2} \quad x-\frac{1}{2}$$

Step-by-Step Solution

Answer

The quotient is \( x^3 + 3x - 1 \) with a remainder of \( 4 \).

Step 1: Set Up Synthetic Division

Begin by writing the divisor, which is \( x - \frac{1}{2} \). In synthetic division, we use the root of the divisor, which is \( \frac{1}{2} \). Write this number on the left side. Next, write the coefficients of the dividend \( x^4 - \frac{1}{2}x^3 + 3x^2 - \frac{5}{2}x + \frac{9}{2} \) in order: \( 1, -\frac{1}{2}, 3, -\frac{5}{2}, \frac{9}{2} \).

Step 2: Perform First Iteration

Bring down the leading coefficient \( 1 \) directly below the bottom line. This \( 1 \) is the starting point of the quotient.

Step 3: Multiply and Add

Multiply the root \( \frac{1}{2} \) by the number that was just brought down, \( 1 \), resulting in \( \frac{1}{2} \). Write this under the next coefficient \( -\frac{1}{2} \), then add them: \(-\frac{1}{2} + \frac{1}{2} = 0 \).

Step 4: Repeat Multiplying and Adding

Now multiply \( \frac{1}{2} \) by the result \( 0 \) from the previous step to get \( 0 \). Write it under the next coefficient, \(3\), and add: \(3 + 0 = 3\).

Step 5: Continue Process

Multiply \( \frac{1}{2} \) by \( 3 \) to get \( \frac{3}{2} \). Write under the next coefficient \(-\frac{5}{2} \) and add: \(-\frac{5}{2} + \frac{3}{2} = -1 \).

Step 6: Final Multiplication and Addition

Multiply \( \frac{1}{2} \) by \(-1 \) to get \(-\frac{1}{2} \). Add this to the last term \( \frac{9}{2} \): \( \frac{9}{2} - \frac{1}{2} = 4 \).

Step 7: Write the Result

The numbers below the line represent the coefficients of the quotient polynomial, and the last number is the remainder. The quotient is \( x^3 + 0x^2 + 3x - 1 \) with a remainder of \( 4 \). Hence, the division results in \( x^3 + 3x - 1 + \frac{4}{x - 1/2} \).

Key Concepts

Polynomial Division

Polynomial division is like long division, but with polynomials instead of numbers. It is used to divide one polynomial by another to get a quotient and a remainder.
In this exercise, we used synthetic division, a shortcut method used specifically when dividing by linear divisors of the form \(x - c\). The main advantage of synthetic division is that it is quicker and less cumbersome than traditional polynomial division, especially for larger polynomials.

• First, the root of the divisor is used – in this case, \(\frac{1}{2}\).
• The coefficients of the polynomial are used, as seen in a compact process instead of rewriting the whole expression.
• The outputs are a quotient and sometimes a remainder, which can be further simplified if necessary.

By applying these steps in synthetic division, we simplify problem-solving in algebra.

Remainder Theorem

The remainder theorem helps us quickly find the remainder of a polynomial division. It states that for a polynomial \( P(x) \) divided by \( x-c \), the remainder of this division is simply \( P(c) \).
In other words, just substitute \( c \) into the polynomial and evaluate it. The result is the remainder without needing to do all the math!

In our synthetic division example, after completing the division, the number left over is \(4\). If we were to evaluate \( P\left(\frac{1}{2}\right) \), we would also find it equals \(4\).

• This theorem is particularly useful in checking our division work.
• It simplifies processes by providing quick and accurate verification.
• Helps in confidently determining whether a certain expression is a factor of the polynomial.

Understanding this theorem broadens our toolkit for handling polynomial problems efficiently.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition or multiplication) that form meaningful mathematical relationships. In this exercise, the polynomial \( x^4 - \frac{1}{2}x^3 + 3x^2 - \frac{5}{2}x + \frac{9}{2} \) is an example of an algebraic expression.

Here are some components of algebraic expressions related to this problem:

• Variables: Symbols like \( x \) that represent numbers.
• Coefficients: Numbers in front of variables that scale them, such as \(1\) and \(\frac{1}{2}\).
• Exponents: Powers that variables are raised to (e.g., the "4" in \(x^4\)).

In polynomial expressions, we manipulate these elements through operations to simplify, factor, calculate, or divide as we did here.

Understanding and managing these components effectively flows into other areas of math, thus enhancing overall proficiency in algebra.