Problem 74 - Algebra and Trigonometry

Question

Factor the expression completely. $$ y^{3}-y^{2}+y-1 $$

Step-by-Step Solution

Answer

The expression factors completely to \((y - 1)(y^2 + 1)\).

Step 1: Identify Grouping Potential

First, observe the expression to determine if there can be any grouping that allows for factoring. The expression is \[y^3 - y^2 + y - 1. \]Notice that we can separate the terms into two groups: \(y^3 - y^2\) and \(y - 1\).

Step 2: Factor by Grouping

Within each group, identify common factors:- In the first group \(y^3 - y^2\), factor out \(y^2\): \[ y^2(y - 1) \]- In the second group \(y - 1\), notice it is already a common factor and can be represented as: \[ 1(y - 1) \]Re-write the expression showing the grouped factors:\[(y^2)(y - 1) + 1(y - 1)\]

Step 3: Factor Out Common Binomial

Since \( (y - 1) \) is common in both terms, factor it out:\[(y - 1)(y^2 + 1) \]Now, the expression is factored into the product of \((y - 1)\) and \((y^2 + 1)\).

Step 4: Check for Further Factorization

Check if \(y^2 + 1\) can be factored further. It is the sum of squares, which cannot be factored further over the real numbers. Thus, the expression is fully factored as \[(y - 1)(y^2 + 1).\]

Step 5: Conclusion

The expression has been factored completely. The final factored form of the expression is \[(y - 1)(y^2 + 1).\]

Key Concepts

Factoring by Grouping

Factoring by grouping is an efficient strategy used when direct factoring is difficult in a polynomial expression. It involves rearranging the terms in a polynomial into smaller groups that can be individually factored, then finding a common factor among them. For instance, consider the polynomial expression \(y^3 - y^2 + y - 1\). Here, we can split the terms into two groups:

• The first group: \(y^3 - y^2\)
• The second group: \(y - 1\)

Within each of these groups, you pinpoint a common factor:

• For \(y^3 - y^2\), \(y^2\) is a common factor.
• \(y - 1\) is already factored in its simplest form.

This allows you to rewrite the expression and spot the common binomial factor of \(y - 1\). The next step is to factor out this binomial, leading to a simplified expression. This method is practical for breaking down complex polynomials into more manageable pieces.

Polynomial Expressions

Polynomial expressions consist of variables raised to whole number powers and coefficients. Understanding their structure is key to factoring. A polynomial like \(y^3 - y^2 + y - 1\) consists of:

• Four terms
• Degrees of the terms, with the highest degree of 3 in this example
• Coefficients, which are the numbers in front of the variables

The key is to look for patterns or groupings of terms that allow factoring. In the given expression, the highest degree (degree of 3) helps identify the leading term, which then influences how you approach other terms. It's essential to recognize common factors and structures, like the difference of squares or trinomial forms, as they guide the way to simplifying the expression. The goal in factoring is always to break down the expression into simpler components that are easier to manage or solve.

Sum of Squares

In mathematics, a 'sum of squares' refers to the addition of two square terms. However, the term \((y^2 + 1)\), found in the final factored expression \((y - 1)(y^2 + 1)\), is a specific form known as a sum of squares. Unlike the difference of squares—which can be factored into real numbers—a sum of squares, such as \((y^2 + 1)\), cannot be factored further over the set of real numbers.This is because there are no two identical binomials that can multiply together to form \((y^2 + 1)\) using only real coefficients. Therefore, when you encounter a sum of squares, it often marks the stopping point in the factoring process. Unless you are working over the complex numbers, where it can be factored as \((y+i)(y-i)\), it's usually simpler to leave it as it is in real-number contexts.Recognizing a sum of squares ensures that you have fully simplified the polynomial as much as possible when factoring real polynomials.