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Chapter 0: Problem 67

31鈥�76 ? Factor the expression completely. $$ y^{3}-3 y^{2}-4 y+12 $$

Short Answer

Expert verified
The completely factored form is \((y - 2)(y + 2)(y - 3)\).

Step by step solution

01

Group the terms

The expression is \( y^3 - 3y^2 - 4y + 12 \). Begin by grouping the first two terms together and the last two terms together: \( (y^3 - 3y^2) + (-4y + 12) \).
02

Factor each group

Factor out the greatest common factor from each group. For \( y^3 - 3y^2 \), factor out \( y^2 \): \( y^2(y - 3) \). For \(-4y + 12\), factor out \(- 4\): \(-4(y - 3) \).
03

Factor out the common binomial

Both groups now have a common binomial factor of \( (y - 3) \). Factor this out of the expression: \((y^2 - 4)(y - 3)\).
04

Factor the difference of squares

The expression \( y^2 - 4 \) is a difference of squares, as \(y^2 - 4 = (y - 2)(y + 2)\). Thus, factor it further: \( (y - 2)(y + 2)(y - 3) \).
05

Write the completely factored expression

Combine all the factored parts to write the completely factored expression: \( (y - 2)(y + 2)(y - 3) \). This is the completely factored form of the original expression.

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Key Concepts

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

Grouping Method
The grouping method is a handy technique in polynomial factoring. It's especially useful when dealing with four-term polynomials. The key idea is to rearrange the terms in pairs and factor them piece by piece.
Let's break it down:
  • Identify and group terms: Divide the polynomial into two groups based on common traits or similar expressions.
  • Factor each group separately: Look for any common factors within each group. This allows you to simplify the polynomial even further.
  • Combine factors: Look for a common factor between the newly factored groups and combine them.
For instance, with the polynomial expression \( y^3 - 3y^2 - 4y + 12 \), grouping results in \((y^3 - 3y^2) + (-4y + 12)\). Here, both groups can be separately factored before being combined with a shared factor. By mastering this method, you'll have a valuable tool for simplifying complex polynomial expressions.
Greatest Common Factor
The Greatest Common Factor (GCF) plays a pivotal role in polynomial factoring. It is the largest factor that divides each term in the group without a remainder, simplifying the polynomial.
To use the GCF in factoring:
  • Identify the GCF: Look at each group of terms and find the highest power of variables and the largest constant that divide each term.
  • Factor out the GCF: Divide each term by the GCF and place the GCF outside the parentheses.
  • Simplify the expression: Rewrite the expression using the new factors.
In the polynomial \( y^3 - 3y^2 - 4y + 12 \), the GCF of the first pair \( y^3 - 3y^2 \) is \( y^2 \), and for \(-4y + 12\), it is \(-4\). Taking out these factors helps streamline the polynomial for further factoring.
Difference of Squares
Understanding the difference of squares is essential for polynomial factoring. A difference of squares is a specific pattern where two terms, each being a square, are subtracted. The formula is: \[ a^2 - b^2 = (a-b)(a+b)\] This form allows for straightforward factoring.
To apply the difference of squares:
  • Recognize the pattern: Ensure the expression fits \( a^2 - b^2 \).
  • Factor accordingly: Write the expression as \( (a-b)(a+b) \).
  • Check your work: Ensure both resulting binomials multiply back to the original expression.
In our example, \( y^2 - 4\) is a difference of squares since \(4\) is \(2^2\). So, it can be broken down further into \( (y - 2)(y + 2) \). Recognizing and using this pattern simplifies polynomial expressions.

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Most popular questions from this chapter

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Degrees of Sums and Products of Polynomials Make up several pairs of polynomials, then calculate the sum and product of each pair. Based on your experiments and observations, answer the following questions. (a) How is the degree of the product related to the degrees of the original polynomials? (b) How is the degree of the sum related to the degrees of the original polynomials? (c) Test your conclusions by finding the sum and product of the following polynomials: $$2 x^{3}+x-3 \quad \text { and } \quad-2 x^{3}-x+7$$

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