Problem 67 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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: \( (y^3 - 3y^2) + (-4y + 12) \).

Step 2: 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 \(- 4\): \(-4(y - 3) \).

Step 3: Factor out the common binomial

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

Step 4: 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: \( (y - 2)(y + 2)(y - 3) \).

Step 5: 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.

Key Concepts

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.