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Chapter 1: Problem 66

Factor the expression by grouping terms. $$x^{5}+x^{4}+x+1$$

Short Answer

Expert verified
The expression factors to \((x^4 + 1)(x + 1)\).

Step by step solution

01

Recognize the expression

The expression to be factored is \(x^5 + x^4 + x + 1\). This polynomial has four terms which suggests that factoring by grouping might be a suitable method.
02

Group terms

Group the terms in pairs that may yield a common factor: \((x^5 + x^4) + (x + 1)\).
03

Factor out the greatest common factor from each group

In the first group \((x^5 + x^4)\), the greatest common factor is \(x^4\), so factor it out: \(x^4(x + 1)\). In the second group \((x + 1)\), the common factor is \(1\), so rewrite it as \(1(x + 1)\). This results in \(x^4(x + 1) + 1(x + 1)\).
04

Factor out the common binomial factor

Now, notice that \((x + 1)\) is common in both grouped terms. Factor \((x + 1)\) out to get \((x^4 + 1)(x + 1)\).
05

Final check

Verify that the factored form \((x^4 + 1)(x + 1)\) expands back to the original expression \(x^5 + x^4 + x + 1\). When expanded, the expression indeed returns to the original.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Expressions
Polynomial expressions are mathematical phrases containing a sum of terms. These terms are made up of variables raised to whole number powers and coefficients. For example, in the expression \(x^5 + x^4 + x + 1\), each term includes a power of \(x\) with a specific coefficient. Here:
  • \(x^5\) has a coefficient of 1.
  • \(x^4\) also has a coefficient of 1.
  • The stand-alone \(x\) has a coefficient of 1 as well.
  • The constant term 1 is essentially \(x^0\) with a coefficient of 1.
Polynomial expressions can be simplified, expanded, and factored using various algebraic techniques. Recognizing and manipulating these expressions is important for solving equations and understanding more advanced mathematical concepts.
Understanding the structure of polynomials helps in identifying when and how to employ different factoring methods like factoring by grouping.
Common Factors
A common factor in a polynomial expression is a number or variable that divides all parts of an expression evenly. Recognizing common factors is an essential skill when working with polynomials.
In the example of \(x^5 + x^4 + x + 1\), we start by grouping the terms to make finding a common factor easier. The terms \(x^5 + x^4\) share a common factor of \(x^4\), while \(x + 1\) does not share any variable factor, but can be considered as having a common factor of 1.
This results in:
  • Grouping \(x^5 + x^4\) as \(x^4(x + 1)\).
  • Grouping \(x + 1\) as \(1(x + 1)\).
Finding a common factor among terms or groups simplifies the algebraic expression by reducing it to its parts.
Factoring Techniques
Factoring is a fundamental process in algebra that reverses the distribution of terms with shared factors. Factoring by grouping is an efficient technique often used when an expression has four terms like \(x^5 + x^4 + x + 1\).
The key stages include:
  • First, split the expression into groups: \((x^5 + x^4) + (x + 1)\).
  • Next, extract the greatest common factor from each group.
  • Identify the common binomial factor \((x+1)\) that appears in each group.
  • Then, factor \((x + 1)\) out of the expression to get \((x^4 + 1)(x + 1)\).
This process allows us to simplify the polynomial into a product of simpler components. Always verify your work by re-expanding the factored form to ensure it matches the original expression. Mastery of these techniques paves the way for tackling more complicated polynomials and mathematical problems.

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Most popular questions from this chapter

The rational expression $$\frac{x^{2}-9}{x-3}$$ is not defined for \(x=3 .\) Complete the tables and determine what value the expression approaches as \(x\) gets closer and closer to \(3 .\) Why is this reasonable? Factor the numerator of the expression and simplify to see why. $$\begin{array}{c|c} \hline x & \frac{x^{2}-9}{x-3} \\ \hline 2.80 & \\ 2.90 & \\ 2.95 & \\ 2.99 & \\ 2.999 & \end{array}$$ $$\begin{array}{|l|l|} \hline x & \frac{x^{2}-9}{x-3} \\ \hline 3.20 & \\ 3.10 & \\ 3.05 & \\ 3.01 & \\ 3.001 & \\ \hline \end{array}$$

Verify these formulas by expanding and simplifying the right-hand side. $$\begin{array}{l}A^{2}-1=(A-1)(A+1) \\\A^{3}-1=(A-1)\left(A^{2}+A+1\right) \\\A^{4}-1=(A-1)\left(A^{3}+A^{2}+A+1\right)\end{array}$$ Based on the pattern displayed in this list, how do you think \(A^{5}-1\) would factor? Verify your conjecture. Now generalize the pattern you have observed to obtain a factoring formula for \(A^{n}-1,\) where \(n\) is a positive integer.

Factor the expression completely. $$y^{4}(y+2)^{3}+y^{5}(y+2)^{4}$$

Factor the expression completely. $$2 x^{2}+5 x+3$$

Factor the expression completely. $$y^{3}-3 y^{2}-4 y+12$$
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