Problem 85 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ \left(x^{2}+2\right)(x+1) $$

Step-by-Step Solution

Answer

Question: Multiply the following algebraic expressions: $(x^2+2)(x+1)$ Answer: $x^3 + x^2 + 2x + 2$

Step 1: Write down the given expression

Given expression: $$(x^2+2)(x+1)$$

Step 2: Use the distributive property

To multiply the expressions, we will apply the distributive property, which states that each term in the first expression, \((x^2+2)\), must be multiplied by each term in the second expression, \((x+1)\). The multiplication will look like this: $$ (x^2+2)(x+1) = x^2(x+1) + 2(x+1) $$

Step 3: Perform the multiplications

Next, we will multiply the terms in each expression: $$ x^2(x+1) + 2(x+1) = x^2\cdot x + x^2\cdot1 + 2\cdot x + 2\cdot1 $$

Step 4: Simplify the expressions

Now, we will simplify each term: $$ x^3+ x^2 + 2x + 2 $$

Step 5: Combine like terms and write the final answer.

In this case, there are no like terms to combine, so the final answer is: $$ x^3 + x^2 + 2x + 2 $$

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (such as addition and multiplication) that together represent a quantity. For example, the expression \(x^2 + 2\) has a variable \(x\), a power of that variable \(x^2\), and a constant number \(2\). These expressions can become very complex and can include multiple terms and variables.

To fully understand algebraic expressions, it's essential to become familiar with principles like the distributive property, which is foundational when you're multiplying expressions. The distributive property allows you to multiply a single term by each term of an expression, enabling you to expand and simplify the expression. In our exercise \( (x^2+2)(x+1) \), the expression is written as a product of two smaller expressions, demonstrating the importance of this concept in algebra.

Multiplying Polynomials Step by Step

Multiplying polynomials involves distributing each term of the first polynomial to every term of the second polynomial, a process sometimes referred to as the FOIL method for binomials. For the exercise \( (x^2+2)(x+1) \) the first term of the first binomial \(x^2\) needs to be multiplied by each term of the second binomial \(x+1\), and then you do the same with the second term of the first binomial, which is \(2\).

Here's an illustration using our example:

• Multiply \(x^2\) by \(x\) to get \(x^3\)
• Multiply \(x^2\) by \(1\) to get \(x^2\)
• Multiply \(2\) by \(x\) to get \(2x\)
• Multiply \(2\) by \(1\) to get \(2\)

When you combine these products, you get the expanded form of the expression before combining like terms.

Combining Like Terms

After expanding an algebraic expression through multiplication, the next step is to combine like terms. Like terms are terms that have the same variable and exponent. They can be added or subtracted from each other to simplify the expression. In the expanded polynomial \(x^3+ x^2 + 2x + 2\), we look for like terms that can be combined. However, in this example, each term is unique in its variables and exponents.

No like terms mean that the expression is already in its simplest form and cannot be simplified further by combining terms. It's still crucial to check for like terms because combining them can further simplify an expression and make it easier to understand or use in subsequent calculations. Being thorough at this stage ensures that you have the simplest form of the expression for any application.