Problem 82 - Algebra and Trigonometry

Question

Factor the expression completely. $$ \left(a^{2}+2 a\right)^{2}-2\left(a^{2}+2 a\right)-3 $$

Step-by-Step Solution

Answer

\((a - 1)(a + 3)(a + 1)^2\)

Step 1: Identify the Structure

First, observe that the expression \( \left(a^2 + 2a\right)^2 - 2\left(a^2 + 2a\right) - 3 \) is a quadratic in form. Let \( u = a^2 + 2a \). Then the expression becomes \( u^2 - 2u - 3 \).

Step 2: Factor the Quadratic Expression

Now we need to factor the quadratic \( u^2 - 2u - 3 \). We are looking for two numbers that multiply to \(-3\) and add to \(-2\). These numbers are \(-3\) and \(1\), so the factorization is \((u - 3)(u + 1)\).

Step 3: Substitute Back in Terms of \( a \)

Now, substitute back \( u = a^2 + 2a \) into the factors. So, \((u - 3)(u + 1)\) becomes \((a^2 + 2a - 3)(a^2 + 2a + 1)\).

Step 4: Factor Each Expression

Factor \( a^2 + 2a - 3 \) by solving for two numbers multiplying to \(-3\) and adding to \(2\) - these are \(-1\) and \(3\). So \((a - 1)(a + 3)\). Then factor \( a^2 + 2a + 1 \) as \((a+1)(a+1)\) because it is a perfect square trinomial.

Step 5: Write the Final Factored Form

Combine all the factored terms: The complete factorization of the original expression is \((a - 1)(a + 3)(a + 1)^2\).

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of the second degree. This means the equation has terms up to the power of two. More precisely, a standard quadratic equation takes the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero.
Quadratic equations often appear in various areas of algebra and are typically solved using methods like factoring, completing the square, or using the quadratic formula.
A key aspect of solving quadratic equations is finding their roots. These are the values of \( x \) for which the equation equals zero.

• The roots can be found by rewriting the quadratic into its factored form.
• Once in factored form, each factor set to zero represents a potential solution.

Understanding the structure of a quadratic equation can simplify the process of finding these solutions.

Substitution Method

The substitution method is a helpful algebraic technique used to simplify complex equations by introducing a substitution variable. This method allows you to temporarily replace part of the equation with a single variable. It simplifies the equation into a more recognizable form.
In the original exercise, the expression \( \left(a^2 + 2a\right)^2 - 2\left(a^2 + 2a\right) - 3 \) is daunting as it is. By letting \( u = a^2 + 2a \), it transforms into a simpler quadratic form, \( u^2 - 2u - 3 \).
By substituting back to the original terms once the calculation is easier, the substitution method links it back to the variables present in the initial problem.

• This technique reduces the complexity of higher-degree equations.
• It is often used when solving equations involving quadratic-like structures.

This simplification leads to clearer, more manageable steps in solving the equation.

Perfect Square Trinomial

A perfect square trinomial is a specific type of trinomial that can be expressed as the square of a binomial. A trinomial is made up of three terms and will have the form \( a^2 + 2ab + b^2 \), which can be factored into \((a + b)^2\).
In the exercise, we identified \( a^2 + 2a + 1 \) as a perfect square trinomial.
It simplifies to \((a+1)^2\), since it follows the pattern where both the first and the last terms are perfect squares, and the middle term is twice the product of the square roots of the first and last terms.

• The significance of recognizing a perfect square trinomial lies in ease of factorization.
• This knowledge allows the factorization process to be faster and more efficient.

Spotting these trinomials saves time and ensures accuracy in solving algebraic equations.

Factored Form

Factored form refers to expressing a polynomial as a product of its factors, simplifying its complexity. This form is particularly useful as it provides not only an elegant expression of the polynomial but also straightforward solutions for its roots.
The original expression represents this process well. Once it was simplified and recognized as quadratic, it was expressed as the factored form \((a - 1)(a + 3)(a + 1)^2\).
Each of these factors represents a piece of the puzzle, providing insight into where the function will cross the x-axis (the roots).

• Factoring can reveal the multiplicity of roots.
• The factored expression facilitates easier evaluation and graphing of the polynomial function.

Understanding how to express polynomials in factored form is foundational for solving equations, graphing quadratic functions, and even higher-level algebra.