Problem 143 - College Algebra

Question

Exercises \(142-144\) will help you prepare for the material covered in the next section. Use the distributive property to multiply: $$2 x^{4}\left(8 x^{4}+3 x\right)$$

Step-by-Step Solution

Answer

The simplified version of the expression \(2x^4(8x^4+3x)\) using the distributive property is \(16x^8 + 6x^5\).

Step 1: Applying the Distributive Property

Applying the distributive property, which states \(a(b + c) = ab + ac\), to the expression \(2x^4(8x^4+3x)\) means multiple \(2x^4\) with each term inside the parentheses individually, resulting in: \(2x^4 \cdot 8x^4 + 2x^4 \cdot 3x\).

Step 2: Perform the Calculations

Now calculate the operations. When multiplying exponents with the same base, add the powers: \(16x^8 + 6x^5\).

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves multiplying terms with variables raised to various powers. In our exercise, we used the distributive property to effectively multiply these terms.

To understand polynomial multiplication, let's break down the steps:

• Firstly, identify the polynomial terms that need to be multiplied. In this instance, it's the monomial \(2x^4\) and the binomial \((8x^4+3x)\).
• Apply the distributive property to each pair of terms by multiplying the monomial with every term inside the binomial.
• This involves performing a simple multiplication for coefficients (numbers in front of the variables) and an operation with the exponents.

Polynomial multiplication allows us to simplify expressions and solve various algebraic equations more efficiently. It is an essential skill for handling algebraic expressions involving multiple variables.

Exponents

Exponents are a shorthand notation to denote repeated multiplication of the same number or variable. In our exercise, we encountered exponents within a polynomial, such as \(x^4\) and \(x\).

Here’s what you need to know about working with exponents:

• Multiplying Like Bases: When you multiply terms with the same base, you add their exponents. For example, \(x^4 \times x\) becomes \(x^{4+1} = x^5\).
• Exponent Basics: An exponent tells you how many times to multiply the base by itself. For instance, \(2^4\) means multiplying 2 four times, resulting in 16.

Exponent rules simplify computations in algebra by converting repeated multiplication into easier calculations. Understanding how to manipulate exponents aids in multiplying and simplifying polynomial expressions efficiently.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and arithmetic operations. In the exercise, expressions like \(2x^4 (8x^4+3x)\) are manipulated using algebraic rules.

Key aspects of dealing with algebraic expressions involve:

• Understanding terms and coefficients: Terms are individual components separated by plus or minus signs. Coefficients are the numbers that multiply the variables.
• Applying rules like the distributive property to expand or simplify expressions.
• Factoring or combining like terms to achieve concise expressions for further calculations or solving equations.

Grasping how algebraic expressions work is fundamental in math as it opens the doorway to solving equations and real-world problems through symbolic representation.