Problem 59 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ 3 x(2 x+3)(2 x+3) $$

Step-by-Step Solution

Answer

The simplified expression is \(12x^3 + 36x^2 + 27x\).

Step 1: Expand the Expression

First, recognize that the expression is of the form \(3x((2x+3)(2x+3))\). Notice that \((2x+3)(2x+3)\) is a perfect square trinomial. Expand it using the formula for squaring a binomial, which is \((a+b)^2 = a^2 + 2ab + b^2\). Here, \(a = 2x\) and \(b = 3\). Substitute to get: \((2x)^2 + 2(2x)(3) + 3^2\).

Step 2: Simplify the Perfect Square Trinomial

Calculate each term in the perfect square trinomial: - \((2x)^2 = 4x^2\)- \(2(2x)(3) = 12x\)- \(3^2 = 9\)Combine these results to get the expanded form: \(4x^2 + 12x + 9\).

Step 3: Multiply by the Monomial

Now that we have the expanded binomial \(4x^2 + 12x + 9\), multiply it by \(3x\): - \(3x imes 4x^2 = 12x^3\)- \(3x imes 12x = 36x^2\)- \(3x imes 9 = 27x\)Combine these products to obtain the final simplified expression: \(12x^3 + 36x^2 + 27x\).

Key Concepts

Binomial Expansion

When we talk about binomial expansion, we mean expanding an expression that involves two terms, often seen in the format \((a + b)^n\). In our current problem, we're dealing with the expression \((2x + 3)^2\). 
To expand this, we use the formula for squaring a binomial: \((a + b)^2 = a^2 + 2ab + b^2\). This formula states that the square of a binomial is equal to the sum of three terms: the square of the first term, twice the product of the two terms, and the square of the second term.
Plugging these values in, where \(a = 2x\) and \(b = 3\), you see the expanded form is \((2x)^2 + 2(2x)(3) + 3^2\). By calculating each part, we simplify it to get \(4x^2 + 12x + 9\).
This process is integral for handling binomials and helps in breaking down more complicated polynomial expressions.

Perfect Square Trinomial

A perfect square trinomial is a special kind of polynomial that results from squaring a binomial. In simpler terms, when you square a binomial like \((2x + 3)\), you end up with something called a perfect square trinomial.
In our context, the expression \((2x + 3)(2x + 3)\) translates to a perfect square trinomial, \(4x^2 + 12x + 9\). Each part of the trinomial comes from the squaring process of the binomial.
- The first term, \(4x^2\), is obtained by squaring the first part of the binomial, \((2x)^2\). - The middle term, \(12x\), is produced by doubling the product of the two parts: \(2 \times (2x \times 3)\). - The last term, \(9\), results from squaring \(3\) or \(3^2\).
This concept is really helpful because recognizing a binomial square can simplify polynomial operations and calculations.

Monomial Multiplication

Monomial multiplication involves multiplying a single term (in this case, a monomial) by another polynomial.
In our exercise, we take the monomial \(3x\) and multiply it by the polynomial obtained from the binomial expansion, \(4x^2 + 12x + 9\). Here’s a step-by-step guide to understand this:
- Multiply \(3x\) with the first term in the polynomial, \(4x^2\), yielding \(12x^3\). - Next, multiply \(3x\) by the second term, \(12x\), which gives \(36x^2\). - Finally, multiply \(3x\) by the constant \(9\), resulting in \(27x\).
Add all these results together to achieve the final polynomial: \(12x^3 + 36x^2 + 27x\).
This straightforward multiplication technique is essential for solving more complex algebraic expressions, giving you a structured way to handle polynomials.