Problem 25 - Precalculus with Calculus Previews

Question

Use synthetic division to find the quotient \(q(x)\) and remainder \(r\) when \(f(x)\) is divided by the given linear polynomial. $$ f(x)=x^{3}-x^{2}+2 ; x+3 $$

Step-by-Step Solution

Answer

Quotient: \(q(x) = x^2 - 4x + 12\), Remainder: \(-34\).

Step 1: Identify the dividend and divisor

The dividend is the polynomial function given as \(f(x) = x^3 - x^2 + 0x + 2\). The divisor is \(x + 3\). We need to perform synthetic division using \(-3\), which is the root of \(x+3 = 0\).

Step 2: Set up the synthetic division

Write the coefficients of the dividend in a row: \(1\) (from \(x^3\)), \(-1\) (from \(x^2\)), \(0\) (coefficient of \(x\)), and \(2\) (constant term). Place \(-3\) to the left (outside) of your division setup.

Step 3: Perform synthetic division

1. Bring down the leading coefficient \(1\) to the bottom row.2. Multiply \(-3\) by \(1\), getting \(-3\). Place this under the next coefficient, \(-1\).3. Add \(-1\) and \(-3\) to get \(-4\) and write it below.4. Multiply \(-4\) by \(-3\), getting \(12\), and place under the next coefficient, \(0\).5. Add \(0\) and \(12\) to get \(12\), and write below.6. Multiply \(12\) by \(-3\), getting \(-36\), and place under the constant, \(2\).7. Add \(2\) and \(-36\) to get \(-34\), writing it as the remainder.

Step 4: Write the result of synthetic division

The bottom row from left to right gives coefficients of the quotient: \(1\), \(-4\), and \(12\). These correspond to \(x^2\), \(x\), and the constant term, respectively. Thus, the quotient is \(q(x) = x^2 - 4x + 12\) and the remainder is \(-34\).

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another polynomial of lower degree. It is similar to the long division process used with numbers. However, instead of digits, we work with terms in polynomials.
Consider it a systematic way of finding how many times the divisor can "fit" into the dividend, represented in terms of polynomials.

• The dividend is the polynomial being divided.
• The divisor is the polynomial you're dividing by.
• The quotient is the result of the division.
• The remainder is what is "left over" after the division.

In the exercise at hand, we used synthetic division, a streamlined form of polynomial division, specifically designed for dividing polynomials by linear polynomials. It involves using the coefficients of the polynomials and simplifying the division process using algebraic techniques.

Remainder Theorem

The Remainder Theorem is a fundamental concept that links polynomial division with evaluation. It states that for any polynomial \( f(x) \), if you divide it by a linear divisor \( x-a \), the remainder of this division is \( f(a) \).
This theorem is quite practical as it allows us to find the remainder swiftly without fully carrying out the division. In our context, after using synthetic division on \( f(x) \) by \( x+3 \), we obtained a remainder of \(-34\).
If we were to evaluate \( f(-3) \), we could see the theorem in action. Performing this calculation:

• Substitute \(-3\) into \( f(x) = x^3 - x^2 + 0x + 2 \).
• Compute \(( -3 )^3 - ( -3 )^2 + 2\) = \(-27 - 9 + 2 = -34\).

This confirms that the remainder is indeed \(-34\), just as the Remainder Theorem predicts.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (such as +, -, *, /) arranged in a cohesive manner. They are fundamental to algebra and appear in various forms, such as polynomials, rational expressions, and others.
When we talk about polynomials, they are a specific kind of algebraic expression made up of terms of the form \( ax^n \), where \( a \) is a coefficient, \( x \) is the variable, and \( n \) is a non-negative integer.

• Each term of a polynomial is separated by a plus or minus sign.
• The degree of a polynomial is determined by the highest exponent present in the expression.

In our example \( f(x) = x^3 - x^2 + 2 \), it is a third-degree polynomial because the highest power of \( x \) is 3. When we perform operations like synthetic division on polynomials, it's crucial to understand these components and how they interact during the calculation process.