Problem 60 - Elementary Algebra

Question

\(\left(x^{2}+13 x+18\right) \div(x+3)\)

Step-by-Step Solution

Answer

The quotient is \x + 10\ with a remainder of \-12\.

Step 1: Identify the polynomial to be divided

In this exercise, we need to divide the polynomial \((x^{2} + 13x + 18)\) by the binomial \((x + 3)\).

Step 2: Set up the division

Using polynomial long division, set up the division of \((x^{2} + 13x + 18)\) by \((x + 3)\).

Step 3: Divide the leading terms

Divide the leading term of the dividend \(x^{2}\) by the leading term of the divisor \(x\). The result is \(x\).

Step 4: Multiply and subtract

Multiply \(x\) by \((x + 3)\), yielding \(x^{2} + 3x\). Subtract this result from \((x^{2} + 13x + 18)\) to get the new polynomial \((10x + 18)\).

Step 5: Repeat the process

Repeat the division process with the new polynomial \((10x + 18)\). Divide the leading term \10x\ by \x\ to get \10\. Then multiply \10\ by \((x + 3)\), resulting in \10x + 30\). Subtract this from \((10x + 18)\) to get \(-12)\.

Step 6: Write the final result

The quotient is therefore \x + 10\ and the remainder is \-12\. This means \(x^{2} + 13x + 18 = (x + 3)(x + 10) - 12\).

Key Concepts

Polynomial Long Division

Polynomial long division is a method to divide a polynomial by another polynomial of lesser or equal degree. It's similar to the long division process we use with numbers. The goal is to find how many times the divisor can fit into different parts of the dividend, just like traditional long division.

Each step in polynomial long division involves the following:

• Dividing the leading term of the dividend by the leading term of the divisor.
• Multiplying the entire divisor by this result.
• Subtracting this product from the dividend to form a new polynomial.
• Repeating these steps with the new polynomial, onward until a polynomial of a lower degree than the divisor is reached.

By following these steps, you can break down complex polynomials into simpler components, easily manage polynomial division, and identify the quotient and remainder accurately.

Division of Polynomials

The division of polynomials involves finding a quotient and a remainder when one polynomial is divided by another. When performing polynomial division, follow a systematic approach to ensure accurate results.

Here are the steps:

• Identify the given polynomial and set up your division problem. The polynomial you want to divide is called the dividend, and the polynomial you're dividing by is the divisor.
• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the result by the entire divisor and subtract this from the dividend.
• This process creates a new polynomial, which now becomes the dividend in the next step.
• Continue the steps until the degree of the new polynomial (the remainder) is less than the degree of the divisor.

Understanding these steps helps in solving polynomial division problems correctly.

For example, in the problem \(\left ( x^2 + 13x + 18 \right ) \div \left ( x + 3 \right )\), after performing the division, we obtain a quotient of \x + 10\ and a remainder of \-12\.

Remainder Theorem

The Remainder Theorem provides a quick way to determine the remainder when a polynomial is divided by a linear divisor of the form \(x - c\). According to the theorem, if we divide a polynomial \(P(x)\) by \(x - c\), the remainder of this division is \P(c)\.

Here's a simple example: If you have \(P(x) = x^2 + 13x + 18\) and you want to find the remainder when divided by \(x + 3\), plug \-3\ into the polynomial:

• Calculate \P(-3)\.
• This equals \-3^2 + 13\left ( -3 \right ) + 18 = 9 - 39 + 18 = -12\.
• Thus, the remainder is \-12\.

Using the Remainder Theorem can help verify your solutions and provide quick insights into polynomial division problems.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. When working with polynomial division, you'll encounter algebraic expressions frequently.

Here are some key points:

• Terms: Individual parts of an algebraic expression separated by addition or subtraction. For example, in \(x^2 + 13x + 18\), there are three terms: \x^2\, \13x\, and \18\.
• Coefficients: The numerical factor in a term. In \13x\, \13\ is the coefficient.
• Constant term: A term without any variables, such as \18\ in the given polynomial.

Understanding algebraic expressions helps in performing operations like simplification, addition, subtraction, and importantly, polynomial division.

For example, during polynomial long division, you break down and manipulate these terms to simplify the expression into a quotient and remainder, as shown in our division of \(x^2 + 13x + 18\) by \(x+3\).