Problem 28 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity a factor. Find the other factor. $$ 18 x^{3}+20 x, 2 x $$

Step-by-Step Solution

Answer

Question: Given the product \(18x^3+20x\) and one of its factors, \(2x\), find the other factor. Answer: The other factor is \(9x^2+10\).

Step 1: Set up the division

Write the expression with the given product and factor, using a division symbol to find the other factor. $$ \frac{18x^3+20x}{2x} $$

Step 2: Distribute the division across the terms

Divide each term in the numerator by the denominator: $$ \frac{18x^3}{2x}+\frac{20x}{2x} $$

Step 3: Divide each term

Perform the division for each term: $$ 9x^2+10 $$

Step 4: Write down the other factor

Now we have found the other factor: $$ Factor 2 = 9x^2+10 $$ So, the expression \(18x^3+20x\) can be factored as \((2x)(9x^2+10)\).

Key Concepts

Factoring Polynomials

When factoring polynomials, our goal is to express a polynomial as a product of simpler polynomials or numbers. Factoring is like reversing multiplication. You take a product and figure out what multiplied to make it.
Polynomials can often be factored into simpler expressions called factors, much like whole numbers can be factored into smaller numbers. In our example, we factor the polynomial \(18x^3 + 20x\) by finding and dealing with its greatest common factor (GCF) which simplifies the expression.

• First, find the greatest common factor of all terms. In the given polynomial, the GCF is \(2x\).
• Next, divide each term by the GCF to find the other factor. Doing so sub-divides the original polynomial into a simpler expression \(9x^2 + 10\).
• The end result is a factored form which is \((2x)(9x^2 + 10)\).

Factoring helps with simplifying polynomials for easy manipulation or further operations like solving equations. Understanding the factorization process also aids in recognizing polynomial identities and their roots.

Division of Monomials

Division of monomials involves dividing one monomial by another. Monomials are algebraic expressions that consist of one term, and this division simplifies them. It's much like dividing numbers, except it also involves variables.
Let's look at the division of monomials within the polynomial division process:

• Identify coefficients and divide them normally. For example, in the problem \(\frac{18}{2}\), the coefficients 18 and 2 give you 9 upon division.
• Next, divide the variables by applying the rule of exponents: subtract the exponents of like bases. If you have \(\frac{x^3}{x}\), it becomes \(x^{3-1} = x^2\).
• Apply this method individually to each term: \(\frac{18x^3}{2x} = 9x^2\) and \(\frac{20x}{2x} = 10\).

This division allows us to simplify complex expressions into a form that is easier to understand and use. The process strengthens your algebra skills by emphasizing the rules of exponents and dividing coefficients.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They are a fundamental part of algebra and involve performing operations on unknown values called variables.
Understanding algebraic expressions is crucial because they form the building blocks of algebraic equations and functions. Here's a breakdown of their components:

• Terms: These are parts of the expression separated by plus or minus signs. In \(18x^3 + 20x\), "18x^3" and "20x" are terms.
• Coefficients: These are the numbers in front of variables or terms. In our expression, 18 and 20 are coefficients.
• Variables: These are symbols like \(x\) that represent unknown values or quantities.
• Constants: These are numbers without a variable attached. They remain unchanged in expressions.

The ability to manage and manipulate algebraic expressions is crucial in simplifying, evaluating, and solving complex mathematical problems. Recognizing different components of expressions makes factoring, dividing, and further equation solving more attainable.