Question

Factort the expression. (Hint: \(\left.\left(a^{n}\right)^{2}=a^{2 n}\right)\) 5 a^{2 n}-9 a^{n} b^{n}-2 b^{2 n}$

Short answer

The factored form of the expression \(5a^{2n} - 9a^{n}b^{n} - 2b^{2n}\) is \((5a^{n} + 2b^{n})(a^{n} - b^{n})\)

Step-by-step solution

Identify common factors between the terms

Notice that each term of the expression \(5a^{2n} - 9a^{n}b^{n} - 2b^{2n}\) has a factor of either \(a^{n}\) or \(b^{n}\), or both.

Rewrite the expression using properties of exponents

Use the hint given that \((a^{n})^{2} = a^{2n}\) to rewrite the expression as \((a^{n})^{2} \cdot 5 - 9 \cdot a^{n} \cdot b^{n} - (b^{n})^{2} \cdot 2\). Notice that the expression can be seen as a quadratic trinomial where \(a^{n} = x\) and \(b^{n} = y\). Rewriting it yields \(5x^{2} - 9xy - 2y^{2}\).

Factor the quadratic expression

The expression \(5x^{2} - 9xy - 2y^{2}\) is a quadratic trinomial of the form \(ax^{2} + bxy + c^{x^{2}}\). It can be factored into the form \((dx + ey)(fx + gy)\), where \(d, e, f\), and \(g\) are constants that satisfy the conditions \(d \cdot f = a\), \(g \cdot e = c\), and \(d \cdot g + f \cdot e = b\). In our case, the equation can be factored into \((5x + 2y)(x - y)\).

Substitute back the original variables

Finally, substitute back \(a^{n}\) for \(x\) and \(b^{n}\) for \(y\) to obtain the answer in terms of \(a\) and \(b\). This gives \((5a^{n} + 2b^{n})(a^{n} - b^{n})\)

Key concepts

Properties of Exponents

To effectively work with algebraic expressions, particularly when factoring, it is crucial to understand the properties of exponents. These rules govern how exponential expressions are manipulated and simplified. Understanding them will allow you to identify patterns and simplify terms before factoring begins.

For instance, when you have an expression like \(a^n\)^2\, the exponent properties tell you that the powers multiply, resulting in \(a^{2n}\). This rule, known as the power of a power property, is essential when decomposing expressions into more manageable pieces. Similarly, the product of powers property, denoted by \(a^m \cdot a^n = a^{m+n}\), allows you to combine terms with the same base, while the quotient of powers, \(a^m \/ a^n = a^{m-n}\), permits the simplification of divided exponents. Knowing these rules ensures that you can prepare an expression for factoring by rewriting it in its simplest form.

Common Factors

A common factor is a shared algebraic term present in all parts of an expression. In the process of factoring quadratic expressions, identifying common factors simplifies the problem by reducing complexity. Consider an expression like \(5a^{2n} - 9a^nb^n - 2b^{2n}\), the initial step is to observe any factors that occur in each term.

Common factors can include numerical coefficients as well as variables raised to identical or lesser powers. For example, \(a^n\) and \(b^n\) are factors that appear in all terms of the given expression. Extracting these common factors first may transform a complex polynomial into a simpler one that resembles a recognizable pattern, such as a quadratic trinomial. Thus, not only do common factors reduce the expression, but they also pave the way for more advanced factoring techniques.

Quadratic Trinomial

A quadratic trinomial is a second-degree polynomial with three terms, typically in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. To factor such expressions, one looks for two binomials that when multiplied together would produce the original trinomial. The challenge lies in finding the correct combination of terms that satisfy the relationships between the coefficients of \(x^2\), \(x\), and the constant term.

Finding these two binomials is often accomplished by looking for two numbers that multiply to give \(ac\) and add to give \(b\). In the case of the expression \(5x^2 - 9xy - 2y^2\), investigating pairs of factors for \(5\) and \(2\) that combine to \(9\) leads to the factored form \(5x + 2y)(x - y)\). The process of factoring quadratic trinomials is instrumental when we deal with polynomial expressions, enabling us to solve equations and simplify expressions.