Problem 74 - 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

Let's distribute the first term of the binomial, which is \(1\), across the trinomial \((x^2 - 3x + 1)\). This will give:\[1 \cdot x^2 - 1 \cdot 3x + 1 \cdot 1 = x^2 - 3x + 1\]

Step 2: Distribute the Second Term

Now, distribute the second term of the binomial \(2x\) across the trinomial \((x^2 - 3x + 1)\). This yields:\[2x \cdot x^2 - 2x \cdot 3x + 2x \cdot 1 = 2x^3 - 6x^2 + 2x\]

Step 3: Combine Like Terms

Add the expressions from Step 1 and Step 2 together:\[x^2 - 3x + 1 + 2x^3 - 6x^2 + 2x\]Combine like terms:- The \(x^3\) term: \(2x^3\)- The \(x^2\) terms: \(x^2 - 6x^2 = -5x^2\)- The \(x\) terms: \(-3x + 2x = -x\)- The constant: \(+1\)Thus, the simplified expression is:\[2x^3 - 5x^2 - x + 1\]

Step 4: Verify the Solution

Review each operation to ensure they are correctly executed:- Distribution in Step 1 and Step 2 are correct.- Like terms in Step 3 are properly combined.After verification, the expression \(2x^3 - 5x^2 - x + 1\) is confirmed to be the simplified form.

Key Concepts

Distributive Property

When we talk about polynomial multiplication, understanding the distributive property is crucial. This property allows us to multiply a single term by each term inside a set of parentheses. Consider the expression given:

\((1+2x)(x^2-3x+1)\).

Here's how the distributive property simplifies the process:

• First, distribute the first term in the binomial, which is \(1\). Multiply \(1\) by every term in the trinomial \((x^2-3x+1)\), giving us \(x^2 - 3x + 1\).
• Next, distribute the second term, \(2x\), across the trinomial. This gives \(2x \cdot x^2 - 2x \cdot 3x + 2x \cdot 1 = 2x^3 - 6x^2 + 2x\).

By methodically applying the distributive property, we break down complex expressions into manageable calculations. This step is foundational before moving on to combining like terms.

Combining Like Terms

Combining like terms is another essential skill for simplifying expressions. After using the distributive property to expand expressions, we often end up with terms that can be summed or subtracted if they share the same variable and exponent. Take a look at our expanded expression:

\[x^2 - 3x + 1 + 2x^3 - 6x^2 + 2x\]
Here is how to combine like terms effectively:

• Collect terms with matching exponents. For the given expression, these are: \(2x^3\), \(x^2\) and \(-6x^2\), \(-3x\) and \(2x\), and the constant \(1\).
• Perform any addition or subtraction required. So, \(x^2 - 6x^2 = -5x^2\); \(-3x + 2x = -x\).
• Now, arrange the terms in descending order: \(2x^3 - 5x^2 - x + 1\).

Through combining like terms, you consolidate similar components of an expression, which is a vital step in simplifying algebraic expressions.

Simplifying Algebraic Expressions

Simplification in algebra means condensing an expression into its simplest, most readable form. It makes solving or interpreting algebraic equations easier while ensuring calculations are accurate. Starting with our original polynomial multiplication, we applied the distributive property and combined like terms to obtain:

\[2x^3 - 5x^2 - x + 1\].
Breaking down the simplification process into steps:

• Always begin by expanding using the distributive property. This ensures all terms are multiplied accurately.
• Next, scan for and combine like terms. This reduces complexity and clarifies the structure of the expression.
• Finally, arrange the expression in a standard form. Typically, polynomials are written in descending order of their variable powers.

By following these steps diligently, you can transform complex expressions into clear, concise, and simpler forms, making them easier to manipulate in further mathematical work.