Problem 57 - Algebra 2

Question

Factor completely. If the polynomial is not factorable, write prime. \(4 y^{3}+24 y^{2}+36 y\)

Step-by-Step Solution

Answer

The complete factorization is \(4y(y+3)^2\).

Step 1: Find the Greatest Common Factor (GCF)

Look for the greatest common factor of the terms in the polynomial. For the polynomial \(4y^3 + 24y^2 + 36y\), observe that each term has a common factor of \(4y\).

Step 2: Factor out the GCF

Extract the greatest common factor \(4y\) from the polynomial. This gives: \[4y(y^2 + 6y + 9)\].

Step 3: Factor the Remaining Quadratic

Look inside the parenthesis to see if the quadratic \(y^2 + 6y + 9\) can be factored further. This is a perfect square trinomial, which can be written as: \[(y+3)(y+3) = (y+3)^2.\]

Step 4: Write the Complete Factored Form

Combine the factored terms. The polynomial completely factored is: \[4y(y+3)^2.\]

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is a crucial step in factoring polynomials. It is the largest factor shared by all terms in a polynomial. Identifying the GCF simplifies the entire factoring process.

For a polynomial like \(4y^3 + 24y^2 + 36y\), each term can be broken down into its prime factors:

• \(4y^3 = 2^2 \times y \times y \times y\)
• \(24y^2 = 2^3 \times 3 \times y \times y\)
• \(36y = 2^2 \times 3^2 \times y\)

In this example, the GCF is determined by taking the lowest power of all common factors, giving us \(4y\).

By factoring out \(4y\), we dramatically reduce the complexity of the polynomial, facilitating further factoring steps.

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that can be expressed as the square of a binomial. It takes the form of \((a + b)^2\) where \(a^2\), \(2ab\), and \(b^2\) represent the respective components of the trinomial.

In our exercise, the trinomial \(y^2 + 6y + 9\) is indeed a perfect square trinomial. Here's why:

• The first term \(y^2\) is a perfect square, equating to \((y)^2\).
• The last term \(9\) is a perfect square, equating to \((3)^2\).
• The middle term \(6y\) is double the product of \(y\) and \(3\), which matches \(2ab\).

Recognizing a perfect square trinomial can greatly simplify the factoring process, allowing us to rewrite \(y^2 + 6y + 9\) as \((y+3)^2\). This step cuts down calculation time and errors.

Quadratic Factorization

Quadratic Factorization involves breaking down a quadratic polynomial into simpler, multipliable parts. Typically, it entails factoring an expression of the form \(ax^2 + bx + c\).

First, identify if the quadratic is factorable or a perfect square trinomial. In our exercise, since \(y^2 + 6y + 9\) is a perfect square trinomial, it simplifies into \((y+3)(y+3)\).

Here's the systematic approach to quadratic factorization:

• Identify the values of \(a\), \(b\), and \(c\).
• Check if it's a perfect square trinomial.
• If not, use techniques like completing the square or factoring by grouping.

For factorable quadratics, as in this instance, you end up with the polynomial's factors, leading to the completely factored form \(4y(y+3)^2\). Factoring in this method prevents errors and reveals a simpler underlying structure.