Problem 65 - Elementary Algebra

Question

Factor completely. $$ 32 y 3+32 y_{2}-18 y-18 $$

Step-by-Step Solution

Answer

The expression factors to \(2(4y - 3)(4y + 3)(y + 1)\).

Step 1: Group the terms

The given expression is \( 32y^3 + 32y^2 - 18y - 18 \). Start by grouping the terms into two parts: \((32y^3 + 32y^2)\) and \((-18y - 18)\).

Step 2: Factor out the greatest common factor (GCF) for each group

In the first group \(32y^3 + 32y^2\), the GCF is \(32y^2\). Factoring that out gives \(32y^2(y + 1)\). In the second group \(-18y - 18\), the GCF is \(-18\). Factoring that out gives \(-18(y + 1)\).

Step 3: Factor by grouping

Observe that both factored groups contain the \((y + 1)\) term. Factor \((y + 1)\) out from the expression: \(32y^2(y + 1) - 18(y + 1) = (32y^2 - 18)(y + 1)\).

Step 4: Simplify the first factor

Notice within \(32y^2 - 18\), a common factor of 2 exists. Factor 2 out from \(32y^2 - 18\), yielding \(2(16y^2 - 9)\).

Step 5: Factor completely

Recognize \(16y^2 - 9\) as a difference of squares, which factors as \((4y)^2 - 3^2\). Express it as \((4y - 3)(4y + 3)\). Hence, the complete factorization is \(2(4y - 3)(4y + 3)(y + 1)\).

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers. When factoring polynomials, finding the GCF is often the first step.
In the expression \(32y^3 + 32y^2 - 18y - 18\), we look at each group of terms separately:

• For \(32y^3 + 32y^2\), both terms share common factors of \(32\) and \(y^2\). So, the GCF is \(32y^2\).
• For \(-18y - 18\), the terms share \(-18\) as the GCF.

By factoring these out, the expression simplifies into smaller, more manageable groups. Without this step, the polynomial would be much more complex to manage. Factoring the GCF reduces the polynomial to a simpler form, setting the stage for further factorization.

Difference of Squares

The difference of squares is a special case in algebra where an expression such as \(a^2 - b^2\) can be factored into \((a - b)(a + b)\).
In our example, after simplifying the expression to \(32y^2 - 18\), we notice that \(16y^2 - 9\) fits the form of a difference of squares. Observe:

• \(16y^2 = (4y)^2\)
• \(9 = 3^2\)
• Thus, \(16y^2 - 9 = (4y)^2 - 3^2\)

This recognition allows us to factor the expression into \((4y - 3)(4y + 3)\). By spotting the difference of squares, we convert a quadratic expression into a neatly factored product, revealing potential roots or solutions more clearly.

Factor by Grouping

Factor by grouping is a method used when there are four or more terms in a polynomial. We group the terms into pairs or sets that can be separately factored.
To break down \(32y^3 + 32y^2 - 18y - 18\), we initially grouped into \((32y^3 + 32y^2)\) and \((-18y - 18)\).

• For the first group, \(32y^3 + 32y^2\), we factor out the GCF \(32y^2\), resulting in \(32y^2(y + 1)\).
• For the second group, \(-18y - 18\), the GCF is \(-18\), leaving us with \(-18(y + 1)\).
• Notice now both groups share a common factor: \((y + 1)\).

By factoring out \((y + 1)\), we further simplify the expression into \((32y^2 - 18)(y + 1)\). This method is invaluable for organizing complex polynomials into products of simpler binomials or trinomials.