Problem 40 - College Algebra

Question

For the following exercises, multiply the polynomials. $$ (x-1)\left(x^{2}-2 x+1\right) $$

Step-by-Step Solution

Answer

The product of the polynomials is \(x^3 - 3x^2 + 3x - 1\).

Step 1: Distribute the Binomial

To multiply the polynomial \((x-1)\) with \((x^2 - 2x + 1)\), we start by distributing \((x-1)\) to each term in the trinomial. This means individually multiplying \((x-1)\) by each of the terms: \(x^2\), \(-2x\), and \(1\).

Step 2: Multiply the First Term

Multiply the first term \(x\) by each term in the trinomial \((x^2 - 2x + 1)\):1. \(x \times x^2 = x^3\)2. \(x \times (-2x) = -2x^2\)3. \(x \times 1 = x\).

Step 3: Multiply the Second Term

Multiply the second term \(-1\) by each term in the trinomial \((x^2 - 2x + 1)\):1. \(-1 \times x^2 = -x^2\)2. \(-1 \times (-2x) = 2x\)3. \(-1 \times 1 = -1\).

Step 4: Combine Like Terms

Now, combine all your results from the distribution:\(x^3 - 2x^2 + x - x^2 + 2x - 1\).Combine like terms:1. The \(x^2\) terms: \(-2x^2 - x^2 = -3x^2\).2. The \(x\) terms: \(x + 2x = 3x\).This gives us the final polynomial: \(x^3 - 3x^2 + 3x - 1\).

Key Concepts

Binomial Distribution

When working with binomial distribution in the context of polynomial multiplication, it’s important to understand how to distribute each term of the binomial across another polynomial. In our example, the binomial is

• \((x-1)\).

This means you will take each term of the binomial and multiply it by every term in the trinomial

• \((x^2 - 2x + 1)\).

The goal is to ensure each term in the trinomial is effectively multiplied by each term in the binomial. This step is crucial in polynomial multiplication and lays the groundwork for correctly finding the resulting polynomial.

Trinomial

A trinomial is simply a polynomial with three terms. In this exercise, the trinomial we are dealing with is \(x^2 - 2x + 1\). Each of these terms contributes to the formation of the final product when multiplied with \((x-1)\). The three terms in this trinomial

• are \(x^2\), \(-2x\), and \(1\).

When multiplying, make sure that each trinomial term is paired with each binomial term. Understanding the properties of a trinomial will help to distribute correctly and combine terms effectively later in the solution.

Like Terms

In order to simplify polynomial expressions, it is crucial to identify like terms. Like terms are terms that contain the same variables raised to the same powers. For example,

• \(x^2\) and \(-x^2\) are like terms.

In our solution, after distributing the terms and obtaining

• \(x^3 - 2x^2 + x - x^2 + 2x - 1\),

we identified like terms to combine:

• the \(x^2\) terms: \(-2x^2 - x^2 = -3x^2\) and
• the \(x\) terms: \(x + 2x = 3x\).

Combining like terms is essential for simplifying your final answer.

Distributive Property

The distributive property is a key principle in mathematics that allows us to multiply a single term or polynomial by another polynomial. It states that for any numbers or expressions \(a\), \(b\), and \(c\),

• \(a(b + c) = ab + ac\).

Applying this to our example, we distribute the binomial

• \((x-1)\)

across the trinomial

• \((x^2 - 2x + 1)\).

This results in multiplying each term one by one:

• \((x imes x^2), (x imes -2x), (x imes 1)\)
and then
• \((-1 imes x^2), (-1 imes -2x), (-1 imes 1)\).

Understanding the distributive property allows you to efficiently and accurately perform polynomial multiplication, ensuring that you don’t miss any steps or mix up terms.