Problem 57 - Elementary and Intermediate Algebra

Question

Multiply. See Example 6. $$ (x+2)\left(x^{2}-2 x+3\right) $$

Step-by-Step Solution

Answer

The product is \(x^3 - x + 6\).

Step 1: Distribute the first term

Start by distributing the first term, \(x\), from \((x+2)\) to each term inside the second polynomial \((x^2 - 2x + 3)\). Perform these individual multiplications:1. \(x \times x^2 = x^3\)2. \(x \times (-2x) = -2x^2\)3. \(x \times 3 = 3x\)These give us the partial product: \(x^3 - 2x^2 + 3x\).

Step 2: Distribute the second term

Next, distribute the second term, \(2\), from \((x+2)\) to each term inside the second polynomial \((x^2 - 2x + 3)\). Perform these multiplications:1. \(2 \times x^2 = 2x^2\)2. \(2 \times (-2x) = -4x\)3. \(2 \times 3 = 6\)These generate another partial product: \(2x^2 - 4x + 6\).

Step 3: Combine all like terms

Add the results from Step 1 and Step 2. Arrange the terms by their degrees:- Start with \(x^3\) terms: \(x^3\)- Combine \(x^2\) terms: \(-2x^2 + 2x^2 = 0\)- Combine \(x\) terms: \(3x - 4x = -x\)- Finally, add the constant term: \(+6\)This results in the final polynomial: \(x^3 - x + 6\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that you use whenever you multiply a single term by a sum of terms.
The formula generally looks like this:

• For any numbers or expressions, \(a\), \(b\), and \(c\), the distributive property can be written as \(a(b + c) = ab + ac\).

This means you distribute the term outside the parentheses to each term inside the parentheses.
In polynomial multiplication, this property helps us break down complex expressions into simpler parts that can be easily combined later.
When dealing with \((x+2)(x^2 - 2x + 3)\), you apply the distributive property twice:

• First, distribute \(x\) over \(x^2 - 2x + 3\). This results in \(x(x^2)\), \(x(-2x)\), and \(x(3)\).
• Then, distribute \(2\) over \(x^2 - 2x + 3\) yielding another set of products: \(2(x^2)\), \(2(-2x)\), and \(2(3)\).

By continually applying this property, you can transform the multiplication of polynomials into a more manageable form.

Polynomial Terms

Polynomial terms are the building blocks of a polynomial expression.
Each term consists of a coefficient (a number) and a variable raised to a power.
In the expression \((x+2)(x^2 - 2x + 3)\), you see a variety of polynomial terms put together.Each individual term in a polynomial is important because:

• The degree of the term is determined by the exponent of the variable. For example, in \(x^2\), the degree is 2.
• A constant term, such as 6, has a degree of 0 because it doesn’t have any variable.

When multiplying polynomials, each term from one polynomial must be multiplied by each term in the other polynomial.
This results in a wide variety of terms that will need to be combined later based on their degree.

Combining Like Terms

After multiplying polynomials, you often end up with several terms that have the same degree.
"Combining like terms" is the process of adding or subtracting these terms to simplify the expression.
Only terms that have exactly the same variable part can be combined!Here's how it works in our specific example:

• From our example, after distributing and multiplying, you end up with several \(x^2\) terms: \(-2x^2\) from the first distribution, and \(2x^2\) from the second.
• When added together, they cancel each other out, resulting in 0.
• Similarly, \(3x\) and \(-4x\) can be combined to give \(-x\).

By combining like terms, you arrive at the simplest form of the polynomial.
For this exercise, that final simplified expression was \(x^3 - x + 6\).
Mastering the process of combining like terms helps you keep polynomial expressions clear and concise.