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Chapter 0: Problem 14

Factor each polynomial by factoring out the common monomial factor. $$ x^{3}-x^{2}+x $$

Short Answer

Expert verified
The polynomial factors to \( x(x^{2} - x + 1) \).

Step by step solution

01

Identify the common factor

Look at each term in the polynomial to find the greatest common factor (GCF). In the polynomial \( x^{3}-x^{2}+x \), each term has at least an \( x \).
02

Factor out the common monomial

Rewrite the polynomial by factoring out the common monomial factor \( x \). This gives us: \( x(x^{2} - x + 1) \).
03

Verify the factored form

Expand \( x(x^{2} - x + 1) \) to ensure it equals the original expression \( x^{3} - x^{2} + x \). Verification shows: \( x \times x^{2} = x^{3} \), \( -x \times x = -x^{2} \), and \( x \times 1 = x \), confirming the factorization is correct.

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Key Concepts

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

Common Monomial Factor
When factoring polynomials, one of the first steps is to look for the common monomial factor in each term of the polynomial. A monomial is simply a single term consisting of constants and variables. For instance, in the polynomial \( x^{3} - x^{2} + x \), each term contains at least one \( x \). This commonality allows us to factor out \( x \), simplifying the polynomial.
Factoring out this common monomial factor is useful because it reduces the complexity of the polynomial, making it easier to work with. Always look for the highest power of a variable that is common across all terms.
Greatest Common Factor (GCF)
The Greatest Common Factor, or GCF, is the largest factor that can be evenly divided into each term of a polynomial. This concept is critical when factoring any polynomial as it helps simplify the expression.
For instance, in our polynomial \( x^{3} - x^{2} + x \), each term is a multiple of \( x \). Therefore, the GCF is \( x \). To find the GCF:
- List the factors of each term.
- Identify the largest factor common to each term.
After finding the GCF, factor it out from the polynomial. Here, we factor out \( x \), yielding: \( x(x^{2} - x + 1) \). Now, the polynomial is simplified better.
Polynomial Verification
Verifying the factorization of a polynomial ensures that the factored form is correct. This is done by expanding the factored polynomial and comparing it to the original.
In our example, after factoring out \( x \) from \( x^{3} - x^{2} + x \), we get \( x(x^{2} - x + 1) \). To verify, expand this expression:
  • Multiply \( x \) by each term inside the parentheses: \( x \times x^{2} \), resulting in \( x^{3} \).
  • Next, \( x \times -x \), resulting in \( -x^{2} \).
  • Lastly, \( x \times 1 \), resulting in \( x \).
Adding these together yields \( x^{3} - x^{2} + x \), which matches the original polynomial.
This process ensures that no errors were made during factorization.

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Most popular questions from this chapter

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