Problem 73 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The expression factors to \( (y - 3)(y - 2)(y + 2) \).

Step 1: Group the Terms

First, split the expression into groups that can be factored separately. Group the terms as:\( (y^3 - 3y^2) - (4y - 12) \)

Step 2: Factor Out the Greatest Common Factor (GCF) from Each Group

In the first group \( (y^3 - 3y^2) \), the GCF is \( y^2 \). For the second group \( -(4y - 12) \), the GCF is \( -4 \). Factor them out:\( y^2(y - 3) - 4(y - 3) \)

Step 3: Factor by Grouping

Notice that \( (y - 3) \) is a common factor in each group. Factor \( (y - 3) \) out of the entire expression:\( (y - 3)(y^2 - 4) \)

Step 4: Factor the Remaining Quadratic Expression

The expression \( y^2 - 4 \) is a difference of squares. This can be factored using the formula \( a^2 - b^2 = (a - b)(a + b) \):\( y^2 - 4 = (y - 2)(y + 2) \)

Step 5: Write the Complete Factorization

Substitute the factorization of \( y^2 - 4 \) back into the expression:\( (y - 3)(y - 2)(y + 2) \)

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest number or expression that can evenly divide multiple terms. When factoring polynomials, it's crucial to begin by identifying the GCF of the polynomial's terms, as it simplifies the process significantly. For instance, consider the expression:

• In the group \(y^3 - 3y^2\), the GCF is \(y^2\) because \(y^2\) is the highest power of \(y\) that can be factored from both terms.
• In the group \(-4y + 12\), the GCF is \(-4\). We factor out the negative to simplify the expression positively, which results in \(-4(y - 3)\).

By determining and factoring out the GCF from each group, the expression's complexity decreases significantly. This step prepares the polynomial for further factoring, ultimately leading it to simpler factors. Recognizing the GCF is an essential skill that emerges from practice and helps in breaking down more complex expressions.

Difference of Squares

The "Difference of Squares" is a common and straightforward factoring technique used in polynomial expressions. It's applicable when you have an expression of the form \(a^2 - b^2\). The difference of squares formula is:

• \(a^2 - b^2 = (a - b)(a + b)\)

In the exercise provided, after using the grouping method and factoring out the GCFs, we're left with the expression \(y^2 - 4\). Observing this expression, we notice:

• \(y^2\) is the square of \(y\), and \(4\) is the square of \(2\).

Thus, \(y^2 - 4\) fits perfectly into the difference of squares form because:

• \(y^2 - 4 = (y - 2)(y + 2)\)

Recognizing and applying the difference of squares allows quick and efficient polynomial factorization, revealing simpler linear factors.

Grouping Method

The Grouping Method is a reliable approach used to factor polynomials that have more than three terms, where traditional methods don't apply easily. The process includes dividing the polynomial into sets of terms that can be factored separately. Here's a closer look:

• The expression \(y^3 - 3y^2 - 4y + 12\) can be divided into two groups, \((y^3 - 3y^2)\) and \((-4y + 12)\).
• Within each group, identify the GCF and factor it out: for \(y^3 - 3y^2\), the GCF is \(y^2\), and for \(-4y + 12\), it's \(-4\).
• This results in \(y^2(y - 3) - 4(y - 3)\).
• Notice that \(y - 3\) appears in both groups, which then factors the entire expression to \((y - 3)(y^2 - 4)\).

The final step requires further factoring of \(y^2 - 4\) by identifying it as a difference of squares. By using the grouping method, students can manage more complex polynomials efficiently, breaking them into simpler parts for easier factorization.