Problem 76 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ \left(a^{2}+2 a\right)^{2}-2\left(a^{2}+2 a\right)-3 $$

Step-by-Step Solution

Answer

The complete factored form is \((a + 3)(a - 1)(a + 1)^{2}\).

Step 1: Identify the Binomial Substitution

The given expression is \( \left(a^{2}+2a\right)^{2} - 2\left(a^{2}+2a\right) - 3 \). Let us set \( x = a^{2} + 2a \). The expression then becomes \( x^{2} - 2x - 3 \).

Step 2: Factor the Quadratic Expression

We need to factor \( x^{2} - 2x - 3 \). We look for two numbers that multiply to \(-3\) (the constant term) and add to \(-2\) (the coefficient of \(x\)). These numbers are \(-3\) and \(1\). Thus, the expression can be factored as \((x - 3)(x + 1)\).

Step 3: Replace the Binomial Substitution

Substitute back \( x = a^{2} + 2a \) into the factored expression. So, \((x - 3)(x + 1)\) becomes \((a^{2} + 2a - 3)(a^{2} + 2a + 1)\).

Step 4: Factor Each Binomial

Factor each binomial separately:1. For \( a^{2} + 2a - 3 \), we find numbers that multiply to \(-3\) and add to \(2\): these are \(3\) and \(-1\). So it factors to \((a + 3)(a - 1)\).2. For \( a^{2} + 2a + 1 \), recognize it as a perfect square trinomial: \((a + 1)^{2}\).

Step 5: Write the Complete Factored Form

Combine the factored expressions from Step 4 to get the fully factored form: \((a + 3)(a - 1)(a + 1)^{2}\).

Key Concepts

Binomial Substitution

Binomial substitution is a handy tool for simplifying complex polynomial expressions. When dealing with an expression like \( \left(a^2 + 2a\right)^2 - 2\left(a^2 + 2a\right) - 3 \), it can be quite tedious. By setting a common part of the expression as a new variable, we make the task simpler. In this exercise, we let \( x = a^2 + 2a \). This substitution transforms our expression into \( x^2 - 2x - 3 \).

Using substitution gives us a cleaner and more manageable quadratic expression. Once we solve it, we simply substitute back the original binomial expression, getting us to the answer more efficiently. This method is particularly helpful with nested polynomials where direct factoring is challenging.

Quadratic Expression

A quadratic expression is a polynomial of degree two. It generally takes the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. Quadratics are common in many math problems because they have straightforward solutions.

For the expression \( x^2 - 2x - 3 \), we are dealing with a quadratic where \( a = 1 \), \( b = -2 \), and \( c = -3 \). Our goal is to factor this into two binomials. To achieve that, we look for two numbers whose product is \( -3 \) (the constant term \( c \)) and whose sum matches \( -2 \) (the middle term \( b \)).

These numbers are \(-3\) and \(1\). Thus, \( x^2 - 2x - 3 \) factors neatly into \( (x - 3)(x + 1) \). Simplifying quadratics is foundational as it aids in sketching parabolas, solving quadratic equations and, as in our exercise, simplifying higher-order polynomials.

Perfect Square Trinomial

A perfect square trinomial is a special kind of quadratic expression. It describes a trinomial that can be expressed as the square of a binomial. Recognizing these helps with fast simplification.

In our case, one of the factored expressions, \( a^2 + 2a + 1 \), is a perfect square trinomial. It fits the pattern \( a^2 + 2ab + b^2 = (a + b)^2 \). By observing this pattern, we see that \( a^2 + 2a + 1 \) equals \((a + 1)^2 \).

• Identify the leading coefficient to determine the square base \( a \).
• Double the base to examine if it matches the middle term \( 2a \).
• Finally, ensure the constant term is the square of the base's linear component.

Recognizing and factoring perfect square trinomials is a powerful tool for simplifying solutions in algebra.

Factored Form

The factored form of a polynomial is a way of writing it as a product of simpler expressions. To fully factor a polynomial, break it down to its irreducible factors.

For our exercise, we began by substituting the complex term with \( x \), simplified the resulting quadratic to \((x - 3)(x + 1)\), and then substituted back to get \( (a^2 + 2a - 3)(a^2 + 2a + 1) \). The next step involved factorizing these binomials further:

• \( a^2 + 2a - 3 \) gave us \((a + 3)(a - 1)\).
• \( (a^2 + 2a + 1) \), being a perfect square trinomial, became \((a + 1)^2\).

Thus, the fully factored form is \((a + 3)(a - 1)(a + 1)^2 \). Breaking expressions into their simplest parts elucidates their structure and simplifies further mathematical manipulations.