Problem 82 - Intermediate Algebra

Question

Use synthetic division to perform each division. $$ \frac{2 x^{3}+3 x^{2}-8 x+3}{x+3} $$

Step-by-Step Solution

Answer

Using synthetic division, the quotient is \( 2x^2 - 3x + 1 \), with no remainder.

Step 1: Identify the Dividend and Divisor

The expression given for division is \( \frac{2x^3 + 3x^2 - 8x + 3}{x + 3} \). The dividend is \( 2x^3 + 3x^2 - 8x + 3 \) and the divisor is \( x + 3 \). We will use synthetic division with \( x = -3 \), the root of the divisor \( x + 3 \).

Step 2: Set Up for Synthetic Division

Write down the coefficients of the dividend: \( 2, 3, -8, 3 \). Next, write \( -3 \) to the left to set up the synthetic division.

Step 3: Perform the Synthetic Division

1. Bring down the leading coefficient \( 2 \) as it is.2. Multiply \( 2 \) by \( -3 \) which is \( -6 \) and add to the next coefficient \( 3 \), resulting in \( -3 \).3. Multiply \( -3 \) by \( -3 \), which is \( 9 \), and add to \( -8 \) to get \( 1 \).4. Multiply \( 1 \) by \( -3 \), which is \( -3 \), and add to the constant term \( 3 \) to get \( 0 \).

Step 4: Write the Result

The result from the synthetic division process gives us the coefficients \( 2, -3, 1 \) for the quotient. The remainder is \( 0 \). Thus, the quotient is \( 2x^2 - 3x + 1 \) and the division leaves no remainder.

Key Concepts

Polynomial Division

Polynomial division is a process very similar to long division performed on numbers, but instead, it's applied to polynomials. It allows us to divide a polynomial, called the dividend, by another polynomial, known as the divisor, to find the quotient and remainder.
This technique is particularly useful when dealing with polynomials of higher degrees where the standard arithmetic division becomes complex.

• **Dividend**: The polynomial you want to divide.
• **Divisor**: The polynomial you divide by.
• **Quotient**: The result of the division, not including the remainder.
• **Remainder**: This is what's left over after completing the division.

The main goal is to simplify the expression by removing factors from the dividend, resulting in a quotient that represents the other factor of the original polynomial. In simpler terms, it's like redistributing the 'parts' of the dividend to see what fully fits inside a divisor. Each step of the division process gives insights into the nature of the polynomial's factors.

Remainder Theorem

The Remainder Theorem is a fundamental concept in polynomial algebra. It provides a quick way to find the remainder of a polynomial division without performing the entire division process. According to the theorem, if a polynomial \( f(x) \) is divided by \( x - c \), then the remainder of this division is \( f(c) \).This theorem makes it very convenient to check if a certain value is a root of the polynomial. If, when \( c \) is substituted into the polynomial \( f(x) \), and the result is zero, then \( x - c \) is a factor of the polynomial.
In simpler terms, you plug the value into the polynomial and if you get zero, congratulations! You've found a root.

• **Application**: Quickly find roots of equations or simplify division steps.
• **Verification**: Use it to verify if a division was done correctly.

By understanding and applying the Remainder Theorem, students can save time and effort while solving polynomial equations.

Polynomial Equations

Polynomial equations are equations where the unknown variable, typically \( x \), is raised to various powers, forming a polynomial expression that is set equal to zero. These equations come in various degrees, such as quadratic (degree 2), cubic (degree 3), and so on.The standard form of a polynomial equation looks like \( a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 = 0 \), where \( a_n \) to \( a_0 \) are constants and \( n \) is the degree of the polynomial.

• **Roots**: Solutions to the polynomial equation where the equation equals zero.
• **Degree**: The highest power of the unknown variable in the polynomial.

Solving polynomial equations involves finding the roots, which may require factoring, synthetic division, or applying the Remainder Theorem. These roots represent the x-values at which the polynomial touches or crosses the x-axis on a graph. Understanding polynomial equations and their solutions gives deeper insights into mathematical modeling and predictions in algebra.