Problem 23 - Elementary and Intermediate Algebra

Question

Factor each polynomial. $$ 27 z^{3}+12 z^{2}+3 z $$

Step-by-Step Solution

Answer

The polynomial is factored as \(3z(9z^2 + 4z + 1)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, observe the coefficients and the variables in each term: 27, 12, and 3. The numbers have a GCF of 3. Additionally, each term contains the variable `z` with the smallest power being 1. Therefore, the GCF of the polynomial is 3z.

Step 2: Factor Out the GCF

Take the GCF, 3z, out of each term of the polynomial:\[ 27z^3 + 12z^2 + 3z = 3z(9z^2) + 3z(4z) + 3z(1). \]After factoring out 3z from each term, you get: \[ 3z(9z^2 + 4z + 1). \]

Step 3: Check for Further Factoring

Examine the polynomial \(9z^2 + 4z + 1\) to check if it can be factored further. Notice that this is a quadratic in the form \(az^2 + bz + c\). Use the discriminant method to determine if it can be factored. Calculate the discriminant: \(b^2 - 4ac = 4^2 - 4(9)(1) = 16 - 36 = -20\). The discriminant is negative, meaning the quadratic cannot be factored further using real numbers.

Step 4: Present the Final Factored Form

Since the inner quadratic \(9z^2 + 4z + 1\) cannot be factored further with real numbers, the complete factored form of the polynomial is \(3z(9z^2 + 4z + 1)\).

Key Concepts

Greatest Common Factor

Finding the Greatest Common Factor (GCF) is like finding the largest puzzle piece that fits into every part of the picture. Think of it as the biggest number (and set of variables) that can be evenly divided into each term of the polynomial. In our exercise, the polynomial is made of terms 27z³, 12z², and 3z. All these coefficients—27, 12, and 3—can be divided by the largest common number, which is 3.

However, there is more! Each term also has the variable 'z'. To find the GCF including variables, select the lowest power of 'z' present in each term. The smallest power here is z¹.

Thus, the greatest common factor for this polynomial is actually the product of these two, which is 3z. With this GCF, we can simplify the polynomial by factoring it out from each term, making future steps much easier.

• Revisit each term: 27z³, 12z², 3z
• Find GCF of numbers: 3
• Find lowest power of z: z¹
• Combine: GCF is 3z

Quadratic Polynomials

Quadratic polynomials are a special kind with the highest degree of 2. They often look like: \( az^2 + bz + c \). These polynomials are essential because they form the foundation for understanding more complex algebraic expressions.

In our exercise, after factoring the GCF, we were left with a quadratic polynomial inside the parentheses: \( 9z^2 + 4z + 1 \). A nice thing about quadratics is that they can often be simplified further into linear factors, turning a multiplayer expression into a simpler one. But this is not always the case since it depends on how the coefficients and constant term work together.

• Highest degree term is squared: \( z^2 \)
• General form is \( az^2 + bz + c \)
• Look inside the polynomial for further factoring

Recognizing quadratic polynomials is crucial as they often require special methods to factor, especially when exploring real solutions. They give a significant lead towards understanding broader algebraic structures.

Discriminant Method

When it comes to understanding if a quadratic polynomial can be factored further using real numbers, the discriminant is a key tool. The discriminant provides insight into the nature of the roots (solutions) of a quadratic equation, given in the formula \(b^2 - 4ac\). If you're solving \(az^2 + bz + c\), it's the perfect checkpost!

The discriminant works like this:

• If \(b^2 - 4ac > 0\), the polynomial has two distinct real roots, and it can be factored into linear terms with real coefficients.
• If \(b^2 - 4ac = 0\), there is one real root, meaning it can still be factored, but it results in a perfect square.
• If \(b^2 - 4ac < 0\), there are no real roots—just complex ones. Thus, no further factoring is possible using real numbers.

In our current example, the discriminant method reveals a value of \(-20\) (from \(4^2 - 4(9)(1)\)), clearly less than zero. This negative result informs us that further factorization with real numbers isn’t possible. Always check the discriminant before proceeding with quadratic factorization—it saves time and understanding!
Using these methods not only helps solve the problem but also nurtures a deeper understanding of how polynomials can beautifully unfold into simpler pieces.