Problem 28 - Introductory Algebra for College Students

Question

Use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$9 x^{2}+3 x+2$$

Step-by-Step Solution

Answer

The trinomial \(9x^{2}+3x+2\) is prime and cannot be factored using real numbers.

Step 1: Analyzing the Trinomial

Here, the trinomial given is \(9x^{2}+3x+2\). The coefficients are 9, 3, and 2.

Step 2: Factorizing the Trinomial

We will try to factorize using the quadratic formula method. The quadratic formula is \(-b ± \sqrt{b^{2}-4ac}/2a\). Here, 'a' is the coefficient of \(x^{2}\) which is 9, 'b' is the coefficient of 'x' which is 3 and 'c' is the constant term which is 2. Substituting these values in the formula gives us the roots (or solutions) of the quadratic equation, which correspond to the values of 'x' that make the equation true.

Step 3: Solve for x using the Quadratic Formula

Plugging in the values of a, b and c into the quadratic formula we get: \(x = \frac{-3 ± \sqrt{(3)^{2}-4*(9)*(2)}}{2*9} = \frac{-3 ± \sqrt{-63}}{18}\). Since the value under the square root is negative, the roots of the equation (and therefore the factorization) are complex numbers. Therefore, the trinomial is considered to be prime in the domain of real numbers.

Step 4: Prime Trinomial

As the trinomial cannot be factored into simpler real factors, we conclude that the trinomial is a prime trinomial.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). The roots or solutions of this equation can tell us where the graph of this quadratic intersects the x-axis. They are calculated using:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula breaks down into a couple of components:

• The \'\(b^2 - 4ac\)\' part under the square root is known as the discriminant. It determines the nature of the roots.
• When \(b^2 - 4ac > 0\), the roots are real and distinct.
• When \(b^2 - 4ac = 0\), there is exactly one real root, often called a repeated root.
• When \(b^2 - 4ac < 0\), the roots are complex, meaning they involve imaginary numbers.

In the exercise provided, the discriminant is \(-63\), which is less than zero, indicating the roots are complex. This means the equation does not have real solutions that can factor the trinomial into real numbers.

Complex Numbers

Complex numbers come into play when dealing with quadratic equations that do not have real roots. These numbers extend our number system by including the square root of negative one, denoted as 'i'.

• A complex number has a real part and an imaginary part, typically written as \(a + bi\).
• Here, \(i\) is defined as \(\sqrt{-1}\), which allows us to work with square roots of negative numbers.

In our specific example, the quadratic equation produced a negative discriminant. The operation \(\sqrt{-63}\) results in a complex number since you cannot take the square root of a negative number in the set of real numbers. So, \(\sqrt{-63} = \sqrt{63}i\). This transformation into complex numbers is crucial when working with quadratic equations that seem unfactorable in the real number system.

Prime Polynomial

A prime polynomial is similar to a prime number; it cannot be factored further using polynomials with integer coefficients. When a quadratic trinomial cannot be split into simpler polynomials with real coefficients, it is considered prime.

• In terms of factorization, think of it as trying to "simplify" or "break down" a number into smaller numbers that multiply back to give you the original number.
• However, not every quadratic trinomial can be factored further.

In this exercise, the trinomial \(9x^2 + 3x + 2\) had a negative discriminant, showing no real roots. Therefore, it cannot be factored as a product of two binomials with real coefficients. In the context of real numbers, this makes it a prime polynomial. While it might seem like a dead end, understanding this aspect is crucial for recognizing the limitations and applications of factorization.