Problem 35 - College Algebra with Corequisite Support

Question

For the following exercises, factor the polynomial. \(25 p^{2}-120 m+144\)

Step-by-Step Solution

Answer

The polynomial factors to \((5p-12)^2\).

Step 1: Recognize the Structure

Observe that the given polynomial, \(25p^2 - 120m + 144\), is a quadratic trinomial. To factor it, check if it fits into a special pattern, such as a perfect square trinomial.

Step 2: Check for a Perfect Square Trinomial

For a perfect square trinomial of the form \(a^2 - 2ab + b^2\), we have components: \(25p^2 = (5p)^2\) and \(144 = (12)^2\). Check the middle term: \(-2ab = -2(5p)(12) = -120m\). Since this matches the middle term, it confirms that the trinomial is a perfect square.

Step 3: Write the Polynomial as a Square

Since the polynomial is a perfect square trinomial, we can write it as \((5p - 12)^2\). This means that the expression is equal to \((5p - 12)(5p - 12)\).

Key Concepts

Quadratic Trinomial

A quadratic trinomial is an algebraic expression consisting of three terms. These expressions are typically written in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The quadratic part comes from the fact that the highest power of the variable is 2. This is an important category of polynomials because they often appear in various mathematical problems and are fundamental in algebra.

• First term: This is usually a squared term, like \(a^2\). In our exercise, it's \(25p^2\).
• Middle term: This term involves the variable to the first power, like \(2ab\). In the exercise, it's \(-120m\).
• Last term: This is a constant term, represented by \(c\). Here, it's \(144\).

Quadratic trinomials can be factored using various techniques, such as splitting the middle term, completing the square, or recognizing special patterns like perfect squares. Recognizing and factoring these expressions is crucial for solving equations and simplifying algebraic expressions.

Perfect Square

A perfect square in algebra refers to a number or expression that can be expressed as the square of another number or expression. For example, 9 is a perfect square because it equals \(3^2\). Similarly, a polynomial can be a perfect square if it can be written as the product of two identical binomials.In the case of a trinomial, it is identified as a perfect square trinomial if it fits the form \(a^2 - 2ab + b^2\) or \(a^2 + 2ab + b^2\). Our expression \(25p^2 - 120m + 144\) is a perfect square because:

• The first term \(25p^2\) is a square: \((5p)^2\).
• The last term \(144\) is a square: \((12)^2\).
• The middle term \(-120m\) relates as \(-2 \cdot 5p \cdot 12\).

Because it meets this pattern, it can be rewritten as the square of a binomial: \((5p - 12)^2\). Recognizing and utilizing perfect square trinomials efficiently simplifies the process of factoring and solving algebraic equations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (such as addition and multiplication) that represent mathematical relationships. These expressions form the building blocks of algebra and come in different forms, from simple to complex.Key aspects of algebraic expressions include:

• Terms: These are individual parts separated by + or - signs. For instance, in \(25p^2 - 120m + 144\), there are three terms: \(25p^2\), \(-120m\), and \(144\).
• Coefficients: These numerical factors multiply the variable. In \(25p^2\), 25 is the coefficient of \(p^2\).
• Variables: Symbols like \(p\) and \(m\), which stand for unknown values.

Understanding algebraic expressions is vital because they allow us to model real-world situations mathematically. They lay the groundwork for developing equations and are essential in mathematical problem-solving and reasoning.