Problem 34 - 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. $$ p^{2}-7 p $$

Step-by-Step Solution

Answer

Add \( \frac{49}{4} \); factor as \( \left(p - \frac{7}{2}\right)^2 \).

Step 1: Understand the Concept of Perfect Square Trinomial

A perfect square trinomial takes the form \( (a \, x + b)^2 \), which expands to \( a^2 \, x^2 + 2abx + b^2 \). Our goal is to rewrite the binomial \( p^2 - 7p \) as a trinomial of this form.

Step 2: Identify Components of Perfect Square

In the given binomial \( p^2 - 7p \), we see that \( a = 1 \) and \( 2ab = -7 \). Therefore, we have \( 2 \cdot 1 \cdot b = -7 \). From this, solve for \( b \) to get \( b = -\frac{7}{2} \).

Step 3: Calculate the Constant to Add

To find the constant that completes the square, calculate \( b^2 \), where \( b = -\frac{7}{2} \). Thus, the constant to add is \( \left(-\frac{7}{2}\right)^2 = \frac{49}{4} \).

Step 4: Form the Perfect Square Trinomial

Add \( \frac{49}{4} \) to \( p^2 - 7p \) to obtain the complete trinomial: \( p^2 - 7p + \frac{49}{4} \).

Step 5: Factor the Perfect Square Trinomial

The trinomial \( p^2 - 7p + \frac{49}{4} \) factors into \( \left(p - \frac{7}{2}\right)^2 \). This is because \( (p-b)^2 = p^2 - 2bp + b^2 \), which matches the trinomial structure.

Key Concepts

Understanding Binomials

A binomial is an algebraic expression containing two distinct terms. For example, in the expression \( p^2 - 7p \), we have two terms: \( p^2 \) and \(-7p\). These two terms together form a binomial. Binomials are often seen as part of larger expressions such as trinomials or polynomials.
When dealing with binomials, you might need to transform them. One common transformation is turning a binomial into a perfect square trinomial. This involves finding the right constant to add to the binomial. The purpose of this transformation is to facilitate further algebraic operations, such as factoring. Understanding binomials is the first step in mastering these algebraic concepts. Once you grasp the basics, you can easily manipulate more complex algebraic expressions.

Factoring Trinomials

Factoring is an important concept in algebra that allows you to break down expressions for easier manipulation. In the case of a trinomial, which is an expression with three terms, factoring involves expressing the trinomial as a product of two binomials.
To factor a trinomial, we look for patterns or special forms, such as perfect square trinomials. A trinomial is a perfect square when it can take the form \((a x + b)^2\). When you factor \( p^2 - 7p + \frac{49}{4} \), it becomes \( (p - \frac{7}{2})^2 \). This involves recognizing that the last term is a perfect square and the middle term is twice the product of the square roots of the first and last terms.
Factoring trinomials is a valuable skill that helps simplify equations and solve quadratic expressions. It allows for the separation of algebraic expressions into simpler components, which can be crucial for finding roots or solutions to equations.

Quadratic Expressions and Their Forms

Quadratic expressions are polynomials of degree 2. They generally have the standard form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. Quadratic expressions can take various forms depending on the situation. Sometimes, they appear as trinomials, a three-term expression like \( p^2 - 7p + \frac{49}{4} \).
Recognizing quadratic expressions is essential because they frequently appear in algebra problems. Understanding their properties can help with operations such as solving, factoring, or finding their roots. A common technique with quadratic expressions is completing the square, which can convert a standard quadratic form into a different expression that’s easier to solve or graph.
By mastering quadratic expressions, you enhance your algebra toolkit, allowing you to tackle more complex algebra problems with ease. Quadratic expressions, with their distinctive parabolic graphs and symmetrical properties, are a cornerstone of algebraic studies.