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Chapter 5: Problem 47

Write an equivalent expression by factoring. $$y^{3}-y^{2}+3 y-3$$

Short Answer

Expert verified
(y - 1)(y^{2} + 3)

Step by step solution

01

Title - Group the Terms

First, group the terms in pairs that can be factored separately. This is a common technique to introduce factoring by grouping: y^{3} - y^{2} + 3y - 3 = (y^{3} - y^{2}) + (3y - 3)
02

Title - Factor Each Group

Next, factor out the greatest common factor (GCF) from each group. y^{3} - y^{2} = y^{2}(y - 1) 3y - 3 = 3(y - 1) So now the expression looks like: y^{2}(y - 1) + 3(y - 1)
03

Title - Factor Out the Common Binomial

Notice that (y - 1) is a common factor in both terms. You can factor this out: y^{2}(y - 1) + 3(y - 1) = (y - 1)(y^{2} + 3)
04

Title - Simplify

Now, simplify the expression if needed. In this case, the expression is already simplified: (y - 1)(y^{2} + 3)

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Key Concepts

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

grouping terms
Grouping terms is the first step when factoring by grouping. It involves breakinga polynomial expression into smaller pairs or groups, which makes them easier to factor.
For example, in the expression given, $$y^{3}-y^{2}+3y-3$$, we can group as follows:
(y^{3} - y^{2}) + (3y - 3)
This step helps us identify smaller expressions that can be factored independently.
Make sure that the grouped terms show a common factor so you can factor them in the next step.
greatest common factor
The Greatest Common Factor (GCF) is the highest factor that divides terms in a group.
Finding the GCF is crucial for simplifying expressions and factoring further.
In the example from the problem, we find the GCF for each group:
- For grouping (y^{3} - y^{2}), the GCF is y^{2}, so it becomes y^{2}(y - 1).
- For grouping (3y - 3), the GCF is 3, so it becomes 3(y - 1).
Recognizing the GCF makes it easier to handle polynomial expressions.
factoring quadratics
Factoring quadratics involves the process of breaking down a quadratic expressioninto the product of two binomials.
In the provided solution, after finding the GCF of each group, we ended up with:y^{2}(y - 1) + 3(y - 1).
Notice that the terms now have a common binomial factor, (y - 1).
Factoring this common binomial factor out, we get:
(y - 1)(y^{2} + 3)
This shows how we can simplify a more complex quadratic expression.
polynomial expressions
Polynomial expressions are algebraic expressions that include variables and coefficients,combined using addition, subtraction, and multiplication. They can be classified by their degree and the number of terms:
  • Monomial: A single term (e.g., 5x, -3y^{2})
  • Binomial: Two terms (e.g., x + 3, y^{2} - 4)
  • Trinomial: Three terms (e.g., x^2 + 4x + 4)
In our example, $$y^{3} - y^{2} + 3y - 3$$ is a polynomial of degree 3 with four terms.In targeting a specific strategy like factoring by grouping, we are able to transform and simplify polynomial expressions effectively.

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Most popular questions from this chapter

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