Problem 47 - 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. $$2 x^{2}-9 x y+9 y^{2}$$

Step-by-Step Solution

Answer

The trinomial \(2x^{2}-9xy+9y^{2}\) is a prime trinomial and cannot be factored.

Step 1: Identify the Form of the Trinomial

Notice that the given trinomial is in the form \(ax^{2}+bxy+cy^{2}\) where \(a=2\), \(b=-9\), and \(c=9\). This form hints that it could be factored as \((dx+ey)^{2}\), a perfect square trinomial, or a prime trinomial that can't be factored.

Step 2: Check if it's a perfect square trinomial

Before trying to decide whether a trinomial is prime or not, we check if it's a perfect square trinomial. Factoring \(2x^{2}-9xy+9y^{2}\) as a perfect square trinomial should give it the form \((dx+ey)^{2}\), where \(d^{2}=2, e^{2}=9\) and \(2de=-9\). Checking these requirements shows us that \(d=\sqrt{2}\), \(e=3\), and \(2de=-3\sqrt{2}\). This is not equal to -9, so the trinomial is not a perfect square trinomial.

Step 3: Check if it's a prime trinomial

Since the trinomial is not a perfect square, we can check if it's a prime trinomial. A trinomial is prime if it cannot be factored further. We know this trinomial isn't a perfect square trinomial, so we might suspect it to be a prime trinomial. However, not all trinomials that aren't a perfect square are prime. To be certain, let's attempt applying the factoring technique for prime trinomials, which is to look for two numbers that multiply to \(a*c\) and add to \(b\). In this case, we look for two numbers that multiply to \(2*9 = 18\) and add to \(-9\). Since there are no such numbers, we can indeed confirm that the trinomial is prime.

Step 4: Conclusion

Since the given trinomial neither fits the perfect square trinomial factoring form, nor does it factor by finding two numbers that multiply to \(ac\) and sum up to \(b\), we conclude that it is a prime trinomial with no further factoring possible.

Key Concepts

Prime Trinomials

In algebra, recognizing prime trinomials is crucial. A prime trinomial is a polynomial that cannot be factored over the integers, meaning it cannot be rewritten as the product of two binomials with integer coefficients. This occurs when no two numbers can be found that multiply to the product of the first and last coefficients and add to the middle coefficient. It acts as a roadblock in factoring, similar to how prime numbers can't be divided by any number other than one and itself.

• Check for multiplication: For a trinomial in the form \(ax^2 + bxy + cy^2\), find two numbers that multiply to \(ac\) and add to \(b\).
• When no such numbers exist, the trinomial is prime.

Understanding when a trinomial is prime saves time in algebra and allows you to appreciate the structure of polynomials.

Perfect Square Trinomials

Perfect square trinomials are special forms of trinomials. They can be factored into repeated binomials. This means, they take the form of \((dx+ey)^2\). Noticing a perfect square trinomial is not just about the look, but about checking specific conditions.

• The first and last terms should be perfect squares themselves, e.g., \(d^2\) and \(e^2\).
• The middle term should be twice the product of the square roots of the first and last terms: \(2de\).

In the trinomial \(2x^2-9xy+9y^2\), it doesn’t satisfy these conditions completely, as the necessary relationships between \(d\), \(e\), and \(2de\) don't align. Thus it's not a perfect square trinomial. This checking process helps eliminate options when factoring.

FOIL Method

The FOIL Method is a useful tool in algebra for checking your work after factoring, especially with binomials. FOIL stands for: First, Outside, Inside, Last, describing the order to multiply terms in two binomials.

• First: Multiply the first terms of each binomial.
• Outside: Multiply the outer terms of the binomials.
• Inside: Multiply the inner terms.
• Last: Multiply the last terms of each binomial.

After multiplying, combine like terms to simplify the expression. This method provides a way to double-check factored results, especially when you suspect a factoring error. With the exercise on \(2x^2-9xy+9y^2\), if an incorrect factorization were suspected, using FOIL could help confirm or negate it.

Algebra for College Students

Algebra for college students extends beyond simple factoring. It involves understanding deeper mathematical structures and being able to manipulate and interpret various forms of equations effectively. Mastering concepts like factoring trinomials, recognizing when a trinomial is prime, and using methods like FOIL are foundational.

• Breakdown of complicated expressions is essential for simplifying and solving.
• Analyzing the properties of functions and polynomials is often required.
• Real-life applications demonstrate the importance of these concepts and methods.

As you progress in algebra, these skills help with more complex problems and lay the groundwork for calculus and other higher-level math courses. Being comfortable with these basics is vital for anyone serious about using math in their field of study.