Problem 72 - Intermediate Algebra

Question

Use synthetic division to perform each division. $$ \left(x^{2}+13 x+42\right) \div(x+6) $$

Step-by-Step Solution

Answer

The quotient is \(x + 7\).

Step 1: Set Up the Synthetic Division

To set up synthetic division for this problem, write down the coefficients of the polynomial \(x^2 + 13x + 42\). These coefficients are 1, 13, and 42. The divisor is \(x + 6\), so we take the opposite sign and use -6 as the number we will divide by in the synthetic division process.

Step 2: Perform Synthetic Division

Start by bringing down the coefficient 1, which is the first coefficient of \(x^2\). Multiply -6 by 1, and write the result underneath the next coefficient, which is 13. Add 13 and -6 to get 7. Next, multiply -6 by 7 and write the result underneath 42. Add 42 and -42 to get 0. The numbers at the bottom now show 1 and 7 as coefficients, with a remainder of 0.

Step 3: Identify the Quotient

The result of synthetic division is the coefficients we found: 1 and 7, with the remainder being 0. This means the quotient is \(x + 7\). The remainder is 0, confirming the division had no remainder.

Key Concepts

Understanding Polynomials

Polynomials are fundamental expressions in mathematics consisting of variables and coefficients, connected through operations of addition, subtraction, and multiplication. At its most basic form, a polynomial is written as an expression like \(ax^n + bx^{n-1} + cx^{n-2} + \ldots + k \), where:

• \(a, b, c, \ldots, k\) are coefficients,
• and \(n\) denotes the degree of the polynomial, being the highest power of the variable \(x\).

Polynomials can vary in degree and complexity but share certain properties that make them manageable and essential in mathematical analysis.
One such property is that polynomials are closed under addition, subtraction, and multiplication, meaning performing these operations on polynomials will still result in a polynomial. This feature is significant when manipulating expressions in algebraic contexts.

Division of Polynomials

Dividing polynomials involves breaking down a polynomial by another, which can be accomplished using several methods. Key techniques include polynomial long division and synthetic division. Understanding the division operation is crucial for simplifying expressions or solving polynomial equations.
The process typically entails two components:

• The **dividend**, which is the polynomial you are dividing.
• The **divisor**, the polynomial you are dividing by.

To successfully divide, subtract multiples of the divisor from the dividend, just like how you would do in regular long division. This step-by-step subtraction ultimately leads to finding the quotient and perhaps a remainder.
Synthetic division is often preferred for its simplicity, especially when dividing by a linear divisor of the form \(x - c\). By understanding polynomial division, students gain deeper insights into algebraic functions and their behaviors.

Algebraic Long Division vs. Synthetic Division

Algebraic long division is a systematic process similar to arithmetic long division, used for dividing polynomials with any degree or form. In algebraic long division, we:

• Determine how many times the leading term of the divisor can "fit" into the corresponding term of the dividend.
• Multiply the entire divisor by this term and subtract from the dividend.
• Repeat the process with the new polynomial formed after subtraction until all parts are divided.

However, this method can become lengthy with higher-degree polynomials.
In contrast, synthetic division simplifies this process but is limited to divisors of the form \(x - c\). Preceding with synthetic division involves:

• Using only coefficients, thereby reducing complexity.
• Applying successively simple arithmetic operations: multiplication and addition.

This streamlined approach makes synthetic division especially beneficial for quick calculations and scenarios where the divisor fits the specific form necessary for use. By mastering both methods, learners can approach polynomial division problems with confidence and adaptability.