Factoring a Quartic Trinomial: $a^4 - 2a^2 - 15$

Q1. Factor the trinomial completely: $a^4 - 2a^2 - 15$

Background

Topic: Factoring Higher-Degree Polynomials

This question tests your ability to factor a quartic (degree 4) trinomial. The process often involves recognizing patterns and possibly using substitution to reduce the quartic to a quadratic form.

Key Terms and Formulas

• Quartic trinomial: A polynomial with three terms and the highest degree of 4.

• Factoring: Writing a polynomial as a product of simpler polynomials.

• Substitution: Sometimes, letting $x = a^2$ can help reduce the quartic to a quadratic.

Step-by-Step Guidance

1. Notice that $a^4 - 2a^2 - 15$ can be rewritten in terms of $a^2$. Let $x = a^2$, so the expression becomes $x^2 - 2x - 15$.

2. Now, factor $x^2 - 2x - 15$ as you would a quadratic trinomial. Look for two numbers that multiply to $-15$ and add to $-2$.

3. Once you have factored the quadratic in terms of $x$, substitute back $a^2$ for $x$ in each factor.

4. Check if each resulting factor can be further factored (for example, if you have a difference of squares).

Try solving on your own before revealing the answer!