Problem 9 - Basic Technical Mathematics with Calculus

Question

Multiply the numerator and the denominator of each fraction by the given factor and obtain an equivalent fraction. $$\frac{2}{x+3} \quad(\text { by } x-2)$$

Step-by-Step Solution

Answer

Equivalent fraction is \(\frac{2x - 4}{x^2 + x - 6}\).

Step 1: Identify the Factor

The problem asks for the fraction \(\frac{2}{x+3}\) to be multiplied by \(x-2\) in both the numerator and the denominator. This factor is \(x-2\).

Step 2: Multiply the Numerator

Multiply the numerator of the fraction by the given factor: \[2 \times (x-2) = 2(x-2)\] This simplifies to \(2x - 4\).

Step 3: Multiply the Denominator

Multiply the denominator of the fraction by the given factor: \[(x+3) \times (x-2)\] Use the distributive property (FOIL method) to simplify:\((x+3)(x-2) = x^2 - 2x + 3x - 6 = x^2 + x - 6\).

Step 4: Write the Equivalent Fraction

Now, combine the results of Step 2 and Step 3 into a single fraction:\[\frac{2x - 4}{x^2 + x - 6}\].

Key Concepts

Equivalent Fractions

The concept of equivalent fractions is essential in algebra and involves multiplying or dividing both the numerator and the denominator of a fraction by the same non-zero value. This transforms the fraction into a form that is numerically different but mathematically the same value.

In the example given, \[\frac{2}{x+3}\] is multiplied by \(x-2\) in both the numerator and the denominator. By doing so, we maintain the original value of the fraction while altering its appearance. This process is crucial, especially when attempting to add, subtract, or compare fractions with different denominators. Remember that an equivalent fraction represents the same part of a whole as the original, just expressed differently.

Distributive Property

The distributive property is a fundamental principle in algebra that allows you to multiply a single term across terms within parentheses. This property states that for any numbers \(a, b,\) and \(c\), \(a(b + c) = ab + ac\).

• This property helps simplify expressions and solve equations more efficiently.
• It's particularly useful in polynomial algebra, where you have to distribute terms over sums or differences within parentheses.

In the given step-by-step solution, we use the distributive property to multiply the term \(x-2\) by each term in the polynomial \(x+3\) during the denominator multiplication, ensuring a more straightforward simplification.

FOIL Method

The FOIL Method is a specific application of the distributive property and is used when multiplying two binomials. FOIL stands for First, Outer, Inner, Last, referring to the order in which you multiply the terms in the binomials.

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outer terms in the product.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms.

When simplifying the denominator \((x+3)(x-2)\), the FOIL method helps you quickly expand it to \(x^2 - 2x + 3x - 6\), which further simplifies to \(x^2 + x - 6\). This method streamlines the process of binomial multiplication.

Polynomial Multiplication

Polynomial multiplication involves multiplying two polynomials together and is an extension of the distributive property. Each term in the first polynomial must be multiplied by each term in the second polynomial.

The process includes:

• Multiplying each term from the first polynomial by each term from the second.
• Combining like terms to simplify the expression.

In our exercise, polynomial multiplication is part of finding the new denominator when we multiply \((x+3)\) by \((x-2)\). This combines both the distributive property and binomial multiplication techniques, helping to form a complete, simplified expression in the solution.