Problem 31 - Algebra A Combined Function

Question

Add the proper constant to each binomial so that the resulting trinomial is a perfect square trinomial. Then factor the trinomial. $$ p^{2}+9 p $$

Step-by-Step Solution

Answer

Add \(\frac{81}{4}\) to form \((p + \frac{9}{2})^2\).

Step 1: Identify the Form

The expression given is a binomial: \(p^{2} + 9p\). To make it a perfect square trinomial, we need to determine what constant should be added. A perfect square trinomial typically has the form \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = p\).

Step 2: Find the Needed Constant

In order to complete the square, we need to find a constant \(b^2\) so that \(2ab = 9p\). Since \(a = p\), we equate \(2pb = 9p\) to find \(b\). Dividing both sides by \(2p\), we find \(b = \frac{9}{2}\). Thus, the constant term we need is \(b^2 = \left(\frac{9}{2}\right)^2 = \frac{81}{4}\).

Step 3: Form the Perfect Square Trinomial

Add \(\frac{81}{4}\) to the binomial to convert it to a perfect square trinomial: \[ p^2 + 9p + \frac{81}{4} \]

Step 4: Factor the Trinomial

The perfect square trinomial \(p^2 + 9p + \frac{81}{4}\) factors as \(\left(p + \frac{9}{2}\right)^2\). This is because when expanded, \(\left(p + \frac{9}{2}\right)^2 = p^2 + 2\cdot p \cdot \frac{9}{2} + \left(\frac{9}{2}\right)^2 = p^2 + 9p + \frac{81}{4}\).

Key Concepts

Understanding Binomials

A binomial is an algebraic expression that consists of exactly two terms. In the original exercise, we have the binomial expression: \( p^2 + 9p \). Each term is separated by a plus or minus sign.
Binomials are quite common in algebra and lay the foundation for creating and understanding more complex polynomial expressions. Here is what composes the binomial in our exercise:

• The first term, \( p^2 \), is a squared term representing a variable raised to the power of two.
• The second term, \( 9p \), includes a linear term with the variable \( p \).

Understanding how these terms interact is crucial for operations like completing the square, which is used to transform a binomial into a perfect square trinomial. By examining the coefficients and terms, you can manipulate binomials effectively in algebra.

Completing the Square

Completing the square is a method used in algebra to convert a quadratic expression into a perfect square trinomial. This process simplifies analysis and can offer a clear pathway to solving equations.
The method involves three essential steps:

• Identify the binomial to be transformed. In this case, \( p^2 + 9p \).
• Determine the constant that will complete the square. You find this by equating the middle term to the product \( 2ab \) from the formula \((a + b)^2 = a^2 + 2ab + b^2\). Here, this leads you to find \( b = \frac{9}{2} \).
• Add the square of your result to form the trinomial. Thus, add \( \left( \frac{9}{2} \right)^2 = \frac{81}{4} \) to the expression, yielding \( p^2 + 9p + \frac{81}{4} \).

The completed trinomial now represents a squared binomial, making it easier to factor and analyze in further calculations.

Factoring Trinomials

Factoring a trinomial is the process of expressing it as the product of two binomials. The perfect square trinomial is an ideal candidate for this, as it fits a specific, easily recognizable pattern.
In our worked example, the trinomial \( p^2 + 9p + \frac{81}{4} \) factors beautifully because it is crafted from the formula \( (a + b)^2 = a^2 + 2ab + b^2 \).

• Recognize it as a perfect square trinomial. This is achieved by confirming the added term \( \frac{81}{4} \).
• Apply the square root of the constant term to write it in the form \( (p + \frac{9}{2})^2 \).
• Expand to verify: \( (p + \frac{9}{2})^2 \) simplifies back to \( p^2 + 9p + \frac{81}{4} \), confirming the factorization.

With practice, factoring trinomials becomes intuitive, allowing easy simplification of polynomials into useful algebraic forms.