Problem 48 - Precalculus Mathematics for Calculus

Question

Perform the indicated operations and simplify. $$(1+2 x)\left(x^{2}-3 x+1\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(2x^3 - 5x^2 - x + 1\).

Step 1: Distribute the First Term

Multiply every term inside the parentheses \((x^{2}-3x+1)\) with the first term \(1\) from \((1+2x)\). This results in: \[1 imes (x^2 - 3x + 1) = x^2 - 3x + 1.\]

Step 2: Distribute the Second Term

Now multiply every term inside the parentheses \((x^2 - 3x + 1)\) with the second term \(2x\) from \((1 + 2x)\). This gives us: \[2x imes (x^2 - 3x + 1) = 2x^3 - 6x^2 + 2x.\]

Step 3: Combine Like Terms

Add the results from Step 1 and Step 2 together: \[x^2 - 3x + 1 + 2x^3 - 6x^2 + 2x.\] Now combine like terms: \[2x^3 + (x^2 - 6x^2) + (-3x + 2x) + 1 = 2x^3 - 5x^2 - x + 1.\]

Step 4: Simplify the Expression

The expression simplifies directly from the previous step and there are no further like terms to combine. Therefore, the simplified expression is: \[2x^3 - 5x^2 - x + 1.\]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to multiply every term inside parentheses by a factor outside the parentheses. This property is expressed mathematically as:

• \(a(b + c) = ab + ac\)

In our exercise, we apply the distributive property to expand the expression \((1 + 2x)(x^2 - 3x + 1)\). First, each term inside \((x^2 - 3x + 1)\) is multiplied by 1, which leaves the expression unchanged: \(x^2 - 3x + 1\).
Next, each term inside the same parentheses is multiplied by \(2x\), resulting in \(2x(x^2 - 3x + 1)\), which further simplifies to \(2x^3 - 6x^2 + 2x\).
The distributive property simplifies complex expressions by breaking them down into more manageable parts, one step at a time, setting the stage for easier addition and combining like terms later.

Combining Like Terms

Combining like terms is another essential algebraic process. Like terms have the same variables raised to the same powers, meaning they can be added or subtracted together. This simplification helps make expressions more concise and easier to manage.
Consider the expression \(x^2 - 3x + 1 + 2x^3 - 6x^2 + 2x\). To combine like terms, group all terms with the same degree or matching variables:

• \(2x^3\) is the only term with a degree of 3.
• \(x^2 - 6x^2\) are like terms with a degree of 2, which combine to \(-5x^2\).
• \(-3x + 2x\) are linear terms (same variables), which combine to \(-x\).
• The constant term stands alone as \(+1\).

Thus, combining these like terms results in \(2x^3 - 5x^2 - x + 1\). This method ensures all possible simplifications are executed before finalizing the expression.

Simplifying Expressions

Simplifying expressions is the process of reducing an algebraic expression to its simplest form. This involves applying operations like the distributive property and combining like terms. In this exercise, the original expression \((1 + 2x)(x^2 - 3x + 1)\) is progressively simplified.
Following the distribution, we arrive at a more complex expression: \(x^2 - 3x + 1 + 2x^3 - 6x^2 + 2x\). However, once the combining of like terms is executed, the simplified form emerges as \(2x^3 - 5x^2 - x + 1\).
This final expression contains no further terms that can be simplified or combined. As a result, this step signals the completion of simplification, offering a neat and complete expression that reflects all the necessary operations. Simplifying expressions not only helps in achieving clarity but also aids in solving equations and inequalities effectively.