Problem 44 - Elementary and Intermediate Algebra

Question

Factor. If a polynomial can't be factored, write "prime." $$ h^{2}-144 $$

Step-by-Step Solution

Answer

The polynomial factors to \((h + 12)(h - 12)\).

Step 1: Recognize the Difference of Squares

The provided polynomial is in the form of a difference of squares: \(h^2 - 144\). In general, a difference of squares is expressed as \(a^2 - b^2\), which can be factored into \((a + b)(a - b)\).

Step 2: Identify the Terms

Identify the terms that are squared. In this polynomial, \(h^2\) is squared, meaning \(a = h\). The second term, \(144\), is \(12^2\), meaning \(b = 12\).

Step 3: Apply the Difference of Squares Formula

Substitute the identified terms into the difference of squares formula: \(a^2 - b^2 = (a + b)(a - b)\). Hence, \(h^2 - 144 = (h + 12)(h - 12)\).

Step 4: Conclusion: Factorization

The factorization of the polynomial \(h^2 - 144\) is \((h + 12)(h - 12)\).

Key Concepts

Difference of Squares

The difference of squares is an important concept in algebra. It refers to expressions of the form \(a^2 - b^2\). This special kind of polynomial is characterized by two square terms separated by a minus sign. For example, in the expression \(h^2 - 144\), \(h^2\) and \(144\) are both squares, and they are joined by a subtraction sign.

To factor a difference of squares, you can use a simple and powerful formula: \(a^2 - b^2 = (a + b)(a - b)\). This means you take the square root of each term, set up a sum and a difference, and then multiply them together.

Understanding this property is fundamental for simplifying and solving various algebra problems. It helps in breaking down complex expressions into manageable factors, making them easier to solve or further manipulate.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into simpler terms or factors that, when multiplied together, produce the original polynomial. Factorization is an essential skill in algebra, as it allows for simplification and solving of polynomial equations.

In the example \(h^2 - 144\), factorization involves recognizing the difference of squares pattern and applying the formula \((a + b)(a - b)\). This leads to the factors \((h + 12)(h - 12)\).

The steps to factor a polynomial are usually:

• Identify any patterns or special forms, such as difference of squares.
• Break down complex polynomials into basic components.
• Use algebraic identities and factorizations to simplify expressions.

Mastering polynomial factorization is useful for solving equations, simplifying expressions, and understanding more advanced mathematical concepts.

Algebraic Expressions

Algebraic expressions are combinations of letters, numbers, and arithmetic operations. They are the building blocks of algebra, representing quantities and allowing for the manipulation of variables.

An expression like \(h^2 - 144\) consists of variables (\(h\)), constants (\(144\)), and operators (subtraction). Understanding algebraic expressions is crucial, as they are used to model and solve real-world problems.

Here are some key aspects of algebraic expressions:

• **Variables:** Letters that represent unknown values. In our case, \(h\) is a variable.
• **Constants:** Fixed values, like \(144\) in the expression.
• **Operations:** Basic arithmetic operations like addition, subtraction, multiplication, and division.

Algebraic expressions serve as the foundation for learning more complex math topics, helping students understand how to efficiently solve equations and factor polynomials.