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Chapter 5: Problem 53

Write an equivalent expression by factoring. $$y^{8}-1-y^{7}+y$$

Short Answer

Expert verified
The equivalent factored expression is \((y - 1)(y^{7} + 1)\).

Step by step solution

01

Group terms to simplify

Rewrite the expression by grouping terms to make it easier to factor. The expression becomes: \[ (y^{8} - y^{7}) + (y - 1) \]
02

Factor out the greatest common factor (GCF) from each group

For the first group \(y^{8} - y^{7}\), the GCF is \(y^{7}\). For the second group \(y - 1\), we can leave it as it is. This results in: \[ y^{7}(y - 1) + 1(y - 1) \]
03

Use the distributive property to factor by grouping

Factor out the common binomial factor \(y - 1\): \[ (y - 1)(y^{7} + 1) \]
04

Confirm the factored expression

Verify that the factored form \((y - 1)(y^{7} + 1)\) expands back to the original expression \(y^{8} - 1 - y^{7} + y\).

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Key Concepts

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

greatest common factor
When factoring polynomials, the greatest common factor (GCF) is like the largest puzzle piece that fits into each term of the polynomial. It鈥檚 the biggest term you can factor out without leaving any fractions. For example, in our problem, we looked at the group \[ y^{8} - y^{7} \]. Here, \[ y^{7} \] is the GCF because it鈥檚 the highest power of y that can be taken out of each term.

Understanding how to find the GCF is essential.

  • First, look at each term separately.
  • Identify the variable or constant that appears in all the terms.
  • The GCF should be the highest power of that variable common in all terms.
By factoring out the GCF, you simplify the polynomial and make the other steps easier to manage.
grouping terms
Grouping is a strategy used to simplify polynomials by making it easier to spot common factors. This involves rearranging and pairing terms in the polynomial. In our example, we started with \[ y^{8} - 1 - y^{7} + y \], and we grouped them into \[ (y^{8} - y^{7}) + (y - 1) \].

To group terms effectively:

  • Look for terms that share a common factor.
  • Rearrange the terms to place these common factors together.
  • Sometimes, you may need to factor out a negative sign for easier grouping.
This technique helps in combining like terms and making the polynomial more manageable for further factoring.
distributive property
The distributive property is a fundamental tool in algebra used to simplify expressions and factor polynomials. It states that \[ a(b + c) = ab + ac \]. In our problem, after grouping and factoring out the GCF, we had \[ y^{7}(y - 1) + 1(y - 1) \]. We used the distributive property to factor by grouping:

  • First, recognize the common binomial factor (in this case, \[ y - 1 \]).
  • Factor \[ y - 1 \] out of each group: \[ y^{7}(y - 1) + 1(y - 1) = (y - 1)(y^{7} + 1) \].
By applying the distributive property, we consolidated the polynomial into its simplified factored form \[ (y - 1)(y^{7} + 1) \], making it easier to work with or solve.

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