Problem 53 - College Algebra

Question

Find each product. $$(2 x+3)^{3}$$

Step-by-Step Solution

Answer

The product of the expression \((2x+3)^3\) is \(8x^3 + 36x^2 + 54x + 27\).

Step 1: Apply Binomial Theorem

Using the Binomial Theorem, expand the expression \( (2x+3)^3 \). This yields: \( (2x)^3 + \binom{3}{1}(2x)^2*3 + \binom{3}{2}(2x)*3^2 + 3^3 \).

Step 2: Simplify each term

Next, simplify each term individually. This results in \(8x^3 + 3*4x^2*3 + 3*2x*9 + 27 \).

Step 3: Continue simplification

Further simplification combines the like terms and results in the final product: \(8x^3 + 36x^2 + 54x + 27 \).

Key Concepts

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, combined using only the operations of addition, subtraction, multiplication, and non-negative whole number exponents. These expressions can have multiple terms, and each term consists of a coefficient joined with a variable raised to an exponent. For example, in the polynomial \(8x^3 + 36x^2 + 54x + 27\), each term contributes to the overall degree of the polynomial.
The degree of a polynomial is determined by the highest power of the variable in the expression. So in this case, the degree is 3 because the first term is \(8x^3\). Understanding polynomials is crucial as they appear frequently in algebra and calculus, representing various mathematical and real-world relationships. They form the basis for equations that we solve to find unknown values.

• Polynomials can be classified by degree, such as linear, quadratic, cubic, etc.
• They are often used in mathematical modeling and problem-solving.

Mastering polynomials allows you to tackle more complex algebraic expressions easily.

Exponents

Exponents are a way to express repeated multiplication of the same number or variable. They are written as a small number to the top right of a base number or variable. In the expression \((2x+3)^3\), the exponent is 3, indicating that \(2x+3\) should be multiplied by itself three times:
\[(2x+3) \times (2x+3) \times (2x+3)\]
Exponents are a concise way to show large products and can greatly simplify algebraic calculations.
Here are a few key points about exponents:

• The base is the number or variable that is being multiplied. In \((2x+3)^3\), \(2x+3\) is the base.
• An exponent of 1 implies the base remains the same: \(x^1 = x\).
• Any base to the zero power is 1, except for zero itself: \(x^0 = 1\).

The laws of exponents include rules for multiplying like bases and dealing with negative or fractional exponents. Understanding these laws helps simplify and manipulate algebraic expressions better.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. They are used to generalize mathematical statements and can range from simple expressions like \(2x+3\) to more complex structures that involve multiple variables and higher powers.
Algebraic expressions are fundamental in formulating equations, inequalities, and various mathematical models. They represent real-life situations where relationships between quantities need to be expressed mathematically.

• They can be categorized into monomials, binomials, and polynomials based on the number of terms they contain.
• Simplifying algebraic expressions is a skill that involves substituting values and reducing expressions to their simplest form.

By mastering algebraic expressions, you gain a powerful tool for solving equations and understanding mathematical principles that appear in many fields, such as physics, engineering, and economics.