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Chapter 4: Problem 2

Determine the following products. $$ (2+a) 4 $$

Short Answer

Expert verified
Question: Find the product of (2+a) and 4. Answer: The product of (2+a) and 4 is 8+4a.

Step by step solution

01

Distribute 4 into the binomial

To distribute 4 into the binomial (2+a), we need to multiply the number 4 by each term in the binomial: $$ 4(2) + 4(a) $$
02

Calculate the product of each term

Now, we need to perform the multiplication for each term: $$ 8 + 4a $$
03

Write the final result

The final result is the sum of the products obtained in Step 2: $$ (2+a) 4 = 8+4a $$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Expression
When we talk about a binomial expression, we're referring to an algebraic expression that consists of exactly two distinct terms. In our exercise, the expression \(2+a\) is the binomial. This means it has two separate parts: the number 2 and the variable \(a\). Binomials are very common in algebra and are often a part of bigger polynomial expressions.

Key characteristics of a binomial include:
  • Having two terms: each term can be a number, a variable, or a product of numbers and variables.
  • Terms are combined using addition or subtraction, as seen in \(2+a\), where addition is used.
Understanding binomials is crucial because they often appear in algebraic operations such as multiplication, adding complexity to problems when combined with other algebraic expressions.
Multiplication
Multiplication is one of the fundamental arithmetic operations, and it extends into the world of algebra by allowing us to multiply algebraic expressions, like binomials, with numbers or other expressions. In the example given, the task is to multiply the binomial \(2+a\) by the number 4.

In this context, multiplication involves applying the distributive property. This means we need to distribute, or "apply," the multiplication to each term within the binomial.
  • First, multiply 4 by the first term, which is 2: \(4 \times 2 = 8\).
  • Then, multiply 4 by the second term, which is \(a\): \(4 \times a = 4a\).
This process ensures that each term in the binomial is multiplied by the number outside, leading to correctly expanded expressions that can simplify algebraic equations.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and sometimes division. They don鈥檛 equate to a specific value but represent a range of values depending on the variables they include. In the exercise, the expression we dealt with was \(8 + 4a\), which is the result of distributing and multiplying the binomial and number given.

Algebraic expressions can vary in complexity and can include:
  • Constants, which are fixed numbers like the 8 in our result.
  • Variables, which are symbols representing unspecified numbers, such as \(a\).
  • Coefficients, which are numbers that multiply the variables, like the 4 in \(4a\).
Knowing how to handle algebraic expressions is crucial as they form the basis for creating and solving equations, inequalities, and much more in mathematics.

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Most popular questions from this chapter

For the following problems, perform the multiplications and combine any like terms. $$ x^{2}(x+5)(x+7) $$

For the following problems, simplify each of the algebraic expressions. $$ 2 k[5 k+3(1+7 k)] $$

For the following problems, perform the multiplications and combine any like terms. $$ 2 a^{2}(a+4)(a+3) $$

Find the product. \((x-8)^{2}\).

For the following problems, perform the multiplications and combine any like terms. $$ (x+3)^{2} $$
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