Problem 24 - College Algebra with Modeling and Visualization

Question

Use grouping to factor the polynomial. \(y^{3}-7 y^{2}+8 y-56\)

Step-by-Step Solution

Answer

The factored form is \((y^2 + 8)(y - 7)\).

Step 1: Group the Terms

First, we will look at the polynomial and group the terms. The given polynomial is \(y^3 - 7y^2 + 8y - 56\). To use the grouping method, we will divide it into two groups: \((y^3 - 7y^2)\) and \((8y - 56)\).

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

Now, identify the GCF for each group.- For the first group \(y^3 - 7y^2\), the GCF is \(y^2\). Factoring this out gives us \(y^2(y - 7)\).- For the second group \(8y - 56\), the GCF is \(8\). Factoring this out gives us \(8(y - 7)\).

Step 3: Combine the Factored Groups

Notice that both terms have a common factor of \((y - 7)\). We can factor \((y - 7)\) out from both groups:\[ y^2(y - 7) + 8(y - 7) = (y^2 + 8)(y - 7) \].

Step 4: Present the Final Factored Form

The polynomial \(y^3 - 7y^2 + 8y - 56\) can be factored into \((y^2 + 8)(y - 7)\) using the method of grouping. This is the simplified factored form.

Key Concepts

Grouping Method

Factoring polynomials using the grouping method is a helpful strategy when you have four terms. It's about organizing these terms into smaller, more manageable groups. Consider a polynomial like \(y^3 - 7y^2 + 8y - 56\). By grouping, you're essentially looking for pairs, or sets, of terms that can be factored similarly.

In this exercise, the terms were grouped into two parts:

• \( (y^3 - 7y^2) \)
• \( (8y - 56) \)

Each group can be made simpler by factoring, which leads to a clearer look at the polynomial.

The key to successful grouping is to separate the polynomial in a way where each set of terms can be factored easily. It might take some practice, but once you get a feel for grouping, it simplifies the polynomial dramatically. As a tip, always look for patterns or common terms within the groups.

Greatest Common Factor

Finding the Greatest Common Factor (GCF) is a crucial step in simplifying polynomials because it reduces each group by the largest factor common to all terms in that group. A GCF makes complex problems more tractable.

For the expression

• \(y^3 - 7y^2\), the GCF is \(y^2\).
• \(8y - 56\), the GCF is \(8\).

Factoring these GCFs out simplifies the expression significantly:

\(y^2(y - 7)\) and \(8(y - 7)\).

This step is vital as it sets the stage for finding common factors between the newly factored groups. Always remember, identifying the GCF reduces the risk of mistakes and leads to a more straightforward polynomial.

Factored Form

The factored form of a polynomial is its expression as a product of simpler polynomials. Achieving this simplified form helps in solving equations, as it reveals roots and makes calculations easier.

In our example, after applying the grouping and factoring the GCF, both parts shared the factor \( (y - 7) \). This commonality allowed us to express the polynomial as: \((y^2 + 8)(y - 7)\).

The factored form \((y^2 + 8)(y - 7)\) is simpler, and it's a critical final step in solving algebraic expressions or equations.

By expressing the polynomial in its factored form, you can evaluate or solve for specific values of variables with more ease. This factorization also facilitates deeper analysis of the polynomial's properties, like identifying its zeros. Understanding factored form brings you a step closer to mastering polynomial equations.