Problem 33 - 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. $$5 x^{2}+2 x+9$$

Step-by-Step Solution

Answer

The trinomial \(5x^{2}+2x+9\) is prime, meaning it can't be factored.

Step 1: Identify a, b, and c

The trinomial \(5x^{2}+2x+9\) is in the form \(ax^2+bx+c\). From the given trinomial, \(a = 5\), \(b = 2\), and \(c = 9\).

Step 2: Finding factors

To factorise, find two numbers that multiply to equal \(ac = a \times c = 45\) (which is \(5 \times 9\)), and add up to \(b = 2\). After attempting to find such numbers, it should become clear that there are no such factors. Hence, it can be inferred that the trinomial is prime, meaning it cannot be factored.

Step 3: Confirmation through FOIL

Since the trinomial is prime and could not be factored, there's no need to use the FOIL method for verification.

Key Concepts

Prime Polynomials

A prime polynomial is like the prime number of algebra. It cannot be factored into simpler polynomials with integer coefficients, other than itself and one. When you're working with trinomials, like the one given in the exercise, determining whether it's prime is crucial.

In solving the trinomial \(5x^2 + 2x + 9\), the attempt to factor by finding two numbers that multiply to 45 (the product of \(a\) and \(c\)) and add to 2 (\(b\)) failed. This indicates that the trinomial is not factorable, hence prime.

Prime polynomials are significant because they represent the simplest form of the expression and cannot be simplified any further. Understanding when a polynomial is prime will save you time and effort in problem-solving.

Quadratic Equations

Quadratic equations are a staple in algebra and appear in the form \(ax^2 + bx + c = 0\). These equations are called quadratic because the highest power of the variable, \(x\), is 2.

The equation you encountered, \(5x^2 + 2x + 9\), is a quadratic expression. However, such expressions can be solved using several methods:

• Factoring: Trying to express the quadratic as a product of two binomials.
• Quadratic Formula: Using the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This is useful if factoring is not straightforward or the polynomial is prime.
• Completing the Square: Rewriting the quadratic to make it easier to solve by taking the square root.

Understanding quadratic equations provides a strong foundation for more complex algebraic concepts. Identifying the coefficients \(a\), \(b\), and \(c\) is the first step in tackling these problems, as shown in the original solution.

FOIL Method

The FOIL Method is popular for expanding two binomials. FOIL stands for First, Outer, Inner, Last, which are the terms you multiply together.

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outer terms.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms in each binomial.

When you see a trinomial, using FOIL can verify a successful factorization. You multiply the corresponding terms and sum them to ensure they recreate the trinomial.

In the exercise, since the trinomial \(5x^2 + 2x + 9\) proved to be prime, the FOIL method could not be applied as there weren't any binomials to multiply. However, understanding this technique is important for tackling problems where factorization is possible.