Problem 38 - Algebra and Trigonometry

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

First, we need to distribute the '1' from the first bracket into the second bracket. This means multiplying each term of the polynomial \(x^2 - 3x + 1\) by 1: \[1(x^2 - 3x + 1) = x^2 - 3x + 1.\]

Step 2: Distribute the Second Term

Next, distribute '2x' from the first bracket into the second bracket. Multiply each term of \(x^2 - 3x + 1\) by \(2x\):\[2x(x^2 - 3x + 1) = 2x^3 - 6x^2 + 2x.\]

Step 3: Combine Like Terms

Now, add up all the terms obtained from the distributions:\[x^2 - 3x + 1 + 2x^3 - 6x^2 + 2x.\] Group them by similar degree: \[2x^3 + (x^2 - 6x^2) + (-3x + 2x) + 1.\]

Step 4: Simplify the Expression

Combine the like terms in each group:- Combine \(x^2 - 6x^2\) to get \(-5x^2\).- Combine \(-3x + 2x\) to get \(-x\).The simplified expression is:\[2x^3 - 5x^2 - x + 1.\]

Key Concepts

Distributive Property

The distributive property is a crucial concept in polynomial multiplication. It involves distributing or multiplying each term in a given expression with every term in another expression. To apply the distributive property in this exercise, we first took each term from the expression \(1 + 2x\) and multiplied it with each term in the polynomial\(x^2 - 3x + 1\). This means:

• 1 multiplied by each term in the second polynomial gives: \(1 \cdot (x^2 - 3x + 1) = x^2 - 3x + 1\)
• 2x multiplied by each term in the second polynomial gives: \(2x \cdot (x^2 - 3x + 1) = 2x^3 - 6x^2 + 2x\)

Each part of the distribution acts as a mini multiplication series, ensuring all combinations of terms are considered. The goal is to ensure no term is left without being paired with another.

Combining Like Terms

Once each term from \(1+2x\) has been distributed across the terms of \(x^2 - 3x + 1\), the next task revolves around combining like terms. Like terms are terms that share the same variable raised to the same power. They can be added together.

• Start by grouping like terms from your expression: \(x^2 + 2x^3 - 6x^2 - 3x + 2x + 1\)
• Terms \(x^2\) and \(-6x^2\) are combined as \(-5x^2\)
• Terms \(-3x\) and \(2x\) are combined as \(-x\)

Combining like terms involves simplifying the expression by adding or subtracting coefficients, not altering the variable expressions themselves. It simplifies the expression while maintaining its overall value.

Algebraic Simplification

After applying the distributive property and combining like terms, the final step requires algebraic simplification. This involves organizing the expression into its simplest form, making it easier to understand and work with.

• After the combining process, the terms are rearranged according to their power: \((2x^3 - 5x^2 - x + 1)\)
• Simplification shows each term clearly and in order of decreasing powers of \(x\)

This step is essential for presenting the expression in a conventional format that informs future operations or evaluations. Simplicity in algebra indicates clarity and efficiency in problem-solving and communication.