Problem 29 - Elementary Algebra

Question

Use a pattern to factor. Check. Identify any prime polynomials. $$ 100 f^{2}-9 h^{2} $$

Step-by-Step Solution

Answer

The factored form is \((10f + 3h)(10f - 3h)\), and both factors are prime.

Step 1: Recognize the difference of squares pattern

Identify that the expression given is a difference of squares. The general form for the difference of squares is: \[a^2 - b^2 = (a + b)(a - b)\]In this expression, we can recognize this pattern by rewriting it as: \[100f^2 - 9h^2\]

Step 2: Write the terms as squares

Rewrite each term as a square. Notice that \(100f^2\) can be written as \((10f)^2\) and \(9h^2\) can be written as \((3h)^2\). This gives us: \[(10f)^2 - (3h)^2\]

Step 3: Apply the difference of squares formula

Apply the difference of squares formula \(a^2 - b^2 = (a + b)(a - b)\) to the expression. Here, \(a = 10f\) and \(b = 3h\). Thus, we factor the expression as: \[(10f + 3h)(10f - 3h)\]

Step 4: Verify if the factors are prime

Check if the factors are prime polynomials. In this case, \((10f + 3h)\) and \((10f - 3h)\) are prime because they cannot be factored further.

Step 5: Verify by expanding

To ensure the factorization is correct, expand the factors back to the original expression: \[(10f + 3h)(10f - 3h) = 100f^2 - (3h)(10f) + (10f)(-3h) - 9h^2 = 100f^2 - 9h^2\]

Key Concepts

polynomial factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors. Factors are polynomials that, when multiplied together, give the original polynomial. Factorization can simplify polynomials and help solve equations. The difference of squares is a common factorization pattern. It means we have something that looks like this: \(a^2 - b^2 = (a + b)(a - b)\). In our example, \(100f^2 - 9h^2\), we identify it as a difference of squares since both terms are perfect squares.

prime polynomial

In mathematics, a prime polynomial cannot be factored further using integer coefficients. For instance, once we factor \(100f^2 - 9h^2\) into \( (10f + 3h)(10f - 3h)\), we check if each factor can be factored more. Here, \(10f + 3h\) and \(10f - 3h\) are prime. They have no common factors other than 1 and cannot be broken down into simpler polynomials.

algebraic patterns

Algebraic patterns are specific forms or arrangements in algebra that help simplify and solve problems. One important pattern is the difference of squares, seen in our problem. By recognizing this pattern, we can factor polynomials quickly. Patterns like \(a^2 - b^2\) help in identifying how to rearrange and factor expressions to make them more manageable. Another example is the square of a binomial \( (a + b)^2 = a^2 + 2ab + b^2 \), which appears often in polynomials.

elementary algebra

Elementary algebra includes basic algebraic concepts like variables, constants, expressions, and equations. It's the foundation for more complex algebra. Concepts like the difference of squares belong here as they introduce fundamental factorization techniques. It also covers understanding how to manipulate expressions and solve simple equations. For example, knowing how to factor \(100f^2 - 9h^2\) into \( (10f + 3h)(10f - 3h)\) requires knowledge from elementary algebra.