Problem 46 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Distribute the First Term

Multiply each term in the polynomial \(2x^2 - x + 1\) by the first term \(x\) in the binomial \((x+1)\). \[x(2x^2 - x + 1) = 2x^3 - x^2 + x\]

Step 2: Distribute the Second Term

Now, multiply each term in the polynomial \(2x^2 - x + 1\) by the second term \(+1\) in the binomial \((x+1)\).\[1(2x^2 - x + 1) = 2x^2 - x + 1\]

Step 3: Combine Like Terms

Add the results from Step 1 and Step 2 together, and then combine like terms.\[2x^3 - x^2 + x + 2x^2 - x + 1 = 2x^3 + ( -x^2 + 2x^2 ) + ( x - x ) + 1\]This simplifies to:\[2x^3 + 1x^2 + 0x + 1 = 2x^3 + x^2 + 1\]

Step 4: Present the Simplified Expression

The simplified result of the operation is:\[2x^3 + x^2 + 1\]

Key Concepts

Binomial Distribution

When dealing with polynomial multiplication involving binomials, binomial distribution is a crucial concept. It's the process of multiplying each term in a polynomial by every term in a binomial. Let's take the expression

• \((x+1)(2x^2-x+1)\)

This task can be broken down into manageable parts using the distributive property.
The distributive property states that \(a(b + c) = ab + ac\), which is used here to distribute each term of the binomial to the polynomial.
In Step 1, multiply the first term \(x\) from the binomial \((x+1)\) with each term in the polynomial \((2x^2-x+1)\), transforming the expression as:

• \[x(2x^2 - x + 1) = 2x^3 - x^2 + x\]

In Step 2, repeat this process with the second term \(+1\) from the binomial:

• \[1(2x^2 - x + 1) = 2x^2 - x + 1\]

The result of distribution is the combination of both operations, allowing you to solve multi-term expressions step by step with clarity.

Combining Like Terms

After distributing, the job isn't quite finished. The next step is to combine like terms to further simplify the expression. Like terms have identical variable parts, meaning their variables and exponents match. For example, the terms \(-x^2\) and \(+2x^2\) can be combined, as they both are \(x^2\) terms. By combining them in this case, the resulting term is

• \(-x^2 + 2x^2 = 1x^2\)

In the exercise, we conduct this operation on all the terms we obtained after distribution:

• \(2x^3\) terms stand alone as there are no other \(x^3\) terms.
• Combine the \(-x^2 + 2x^2\) to get \(x^2\).
• The \(x\) and \(-x\) terms cancel each other out: \(x - x = 0\).

By removing or combining these like terms, we get the neat and simplified form

• \(2x^3 + x^2 + 1\)

It’s crucial to combine like terms correctly to ensure that your polynomial is in its simplest form.

Polynomial Simplification

The final process of simplifying polynomials involves both distributing and combining like terms, as we've seen.
Through this methodology, we minimize the polynomial to its simplest form. The good thing about a simplified polynomial is that it's easier to understand and use in further calculations or applications. A simplified polynomial is one where all like terms have been combined, and all possible arithmetic operations have been completed.
This ensures terms are as concise as possible, reducing redundancy. In our exercise, upon completion, our expression is neatly expressed as

• \(2x^3 + x^2 + 1\)

This means all excess terms have been refined to give a clear polynomial that represents the original, more complex set of operations.
Understanding these steps not only helps in solving math homework but also aids in simplifying other computational tasks involving polynomials.