Problem 40 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

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

Step 1: Expand the Expression

To multiply the polynomials, we'll distribute each term in the first polynomial, \((x - 1)\), by each term in the second polynomial, \((x^2 - 2x + 1)\). Start by expanding using the distributive property: \((x - 1)(x^2 - 2x + 1) = x(x^2 - 2x + 1) - 1(x^2 - 2x + 1)\).

Step 2: Distribute the First Term

First, we distribute \(x\) across each term in \((x^2 - 2x + 1)\):1. \(x \cdot x^2 = x^3\)2. \(x \cdot (-2x) = -2x^2\)3. \(x \cdot 1 = x\)Putting these terms together, we get: \(x^3 - 2x^2 + x\).

Step 3: Distribute the Second Term

Next, distribute \(-1\) across each term in \((x^2 - 2x + 1)\):1. \(-1 \cdot x^2 = -x^2\)2. \(-1 \cdot (-2x) = 2x\)3. \(-1 \cdot 1 = -1\)Putting these terms together, we get: \(-x^2 + 2x - 1\).

Step 4: Combine Like Terms

Now we add the results from the distribution steps together:\[(x^3 - 2x^2 + x) + (-x^2 + 2x - 1) = x^3 - 2x^2 + x - x^2 + 2x - 1\]Combine like terms:- The \(x^2\) terms: \(-2x^2 - x^2 = -3x^2\)- The \(x\) terms: \(x + 2x = 3x\)Thus, the combined expression is: \[x^3 - 3x^2 + 3x - 1\].

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows you to expand expressions in a straightforward manner. It states that multiplying a sum by a number is the same as multiplying each addend by the number, and then adding the products. In simpler terms, given three numbers, \(a\), \(b\), and \(c\), the distributive property is expressed as:

• \(a(b + c) = ab + ac\)

Applying this concept to polynomials makes it possible to multiply them effectively. Let's consider the exercise: to multiply \(x - 1\) with \(x^{2} - 2x + 1\). Here, every term in the first polynomial (\(x - 1\)) is multiplied by each term in the second polynomial (\(x^{2} - 2x + 1\)).
This process is crucially important as it ensures all parts of the polynomial are accounted for in the multiplication process. By applying the distributive property, the exercise breaks down into manageable steps, simplifying the multiplication of each term and making it less error-prone.

Combining Like Terms

After distributing the terms across each other, you will often end up with expressions that have similar terms that can be combined. This is known as "combining like terms." In mathematics, like terms are terms that contain the same variables raised to the same power. For instance, in the expression \(2x^{2} + 3x - x^{2} + 4\), \(2x^{2}\) and \(-x^{2}\)\ are like terms.

• To combine like terms, simply add or subtract the coefficients while keeping the variable and its power unchanged.

In the solution, after fully distributing the polynomials, you are left with different terms that must be simplified. For example, \[x^{3} - 2x^{2} + x - x^{2} + 2x - 1\]. Here, combining like terms results in simplifying the \(x^{2}\) terms \( -2x^{2} - x^{2} = -3x^{2}\) and \(x\) terms \(x + 2x = 3x\). This step is necessary for achieving the simplest form of the polynomial.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and arithmetic operations. Algebraic expressions are the foundation of all algebra and are what you manipulate when solving polynomial equations.

• In our objective of multiplying polynomials, each part of the expression such as \(x-1\) and \(x^{2} - 2x + 1\) is a separate polynomial expression.
• These parts include terms (like \(x^{2}\), \(-2x\), and \(1\)).

Understanding the structure of algebraic expressions helps when applying operations such as the distributive property and combining like terms.
In working through the exercise, it's key to identify each term and handle them accordingly through each step of multiplication and simplification. By mastering algebraic expressions, you strengthen your ability to work with equations, understand functions, and tackle complex mathematical problems with ease.