Question

The expression \(\left(\frac{2}{3} x+1\right)\left(\frac{3}{4} x-1\right)=?\) (A) \(\frac{1}{6} x^{2}-\frac{1}{3} x+1\) (B) \(\frac{1}{4} x^{2}+\frac{1}{12} x-4\) (C) \(\frac{1}{2} x^{2}+\frac{1}{12} x-1\) (D) \(x^{2}+\frac{1}{4} x-1\)

Short answer

Option C: \( \frac{1}{2}x^2 + \frac{1}{12}x - 1 \)

Step-by-step solution

Identify the Expression

The expression we need to solve is given as \( \left(\frac{2}{3}x + 1\right)\left(\frac{3}{4}x - 1\right) \). Our goal is to expand this expression and simplify it.

Expand Using the Distributive Property

Apply the distributive property, sometimes known as the FOIL method for binomials:1. Multiply the first terms: \( \frac{2}{3}x \times \frac{3}{4}x = \frac{2 \times 3}{3 \times 4}x^2 = \frac{1}{2}x^2 \).2. Multiply the outer terms: \( \frac{2}{3}x \times -1 = -\frac{2}{3}x \).3. Multiply the inner terms: \( 1 \times \frac{3}{4}x = \frac{3}{4}x \).4. Multiply the last terms: \( 1 \times -1 = -1 \).

Combine Like Terms

Combine the results of the expanded multiplication:- The expression becomes \( \frac{1}{2}x^2 + \left(-\frac{2}{3}x + \frac{3}{4}x\right) -1 \).- Combine the linear terms: \(-\frac{2}{3}x + \frac{3}{4}x = \frac{-8}{12}x + \frac{9}{12}x = \frac{1}{12}x \).- Therefore, the expression simplifies to \( \frac{1}{2}x^2 + \frac{1}{12}x - 1 \).

Match with Provided Options

Compare the simplified expression \( \frac{1}{2}x^2 + \frac{1}{12}x - 1 \) with the given options:- Option A: \( \frac{1}{6} x^{2}-\frac{1}{3} x+1 \)- Option B: \( \frac{1}{4} x^{2}+\frac{1}{12} x-4 \)- Option C: \( \frac{1}{2} x^{2}+\frac{1}{12} x-1 \)- Option D: \( x^{2}+\frac{1}{4} x-1 \)The matching expression is Option C.

Key concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term across terms inside parentheses. It is expressed as:

• For any numbers or expressions, \( a(b + c) = ab + ac \).

This property is especially useful when expanding expressions, such as binomials. Imagine you have an expression like \((x+y)(a+b)\). To find the expanded form, multiply each term in the first binomial by each term in the second binomial. This process follows the pattern:

• First: Multiply the first terms.
• Outer: Multiply the outer terms.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms.

This pattern is often remembered by the acronym FOIL. In our original exercise, we used it to expand the expression,\(\left(\frac{2}{3}x + 1\right)\left(\frac{3}{4}x - 1\right)\), providing a systematic means of breaking down the multiplication into manageable steps. The distributive property simplifies complex expressions and helps in solving algebra problems effectively.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. In algebra, letters are used to represent numbers, making it possible to express and solve equations.
When working with algebra, understanding how to simplify and manipulate expressions is crucial. In the exercise, we used algebraic manipulation to expand and simplify a binomial expression. Here, concepts like variable distribution, combining like terms, and using arithmetic were all components of the process.Key principles in algebra include:

• Identifying expressions and operations.
• Applying operations such as addition, subtraction, multiplication, and division to simplify expressions.
• Using logical reasoning to deal with equations and inequalities.

Through these principles, algebra provides the tools to break down complex expressions, such as in the binomial expansion of \( (\frac{2}{3}x + 1)(\frac{3}{4}x - 1) \), into simpler parts that can be systematically worked through.

Binomial Expansion

Binomial expansion is the process of multiplying two binomials to get a polynomial. A binomial is simply a polynomial with two terms, and the expansion involves applying the distributive property to multiply each term in one binomial by each term in the other.
The general form for expanding binomials is derived from the binomial theorem, but for simplicity, we use the FOIL method when dealing with binomials consisting of two terms:

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

In the original exercise, this process helped us find the product of \(\left(\frac{2}{3}x + 1\right)\left(\frac{3}{4}x - 1\right)\) to produce a polynomial. We combined terms to simplify the expression into\(\frac{1}{2}x^2 + \frac{1}{12}x - 1\).Binomial expansion makes complex algebraic tasks manageable by turning them into orderly sequences of simpler calculations.