Problem 44 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 25 d^{2}-64 f^{2} $$

Step-by-Step Solution

Answer

The fully factored form is .

Step 1: Identify the expression form

Observe that the given polynomial is in the form of a difference of squares, which can be identified as follows:

Step 2: Apply the difference of squares formula

Recall that the difference of squares formula is: . Using this formula, factor the given polynomial: .

Step 3: Combine the factors

Combine the results from Step 2 to get the factorized form of the polynomial: .

Step 4: Verify the factorization

Multiply the factors to check if they simplify back to the original polynomial: .The factorization is verified and the polynomial is fully factored.

Key Concepts

difference of squares

The difference of squares is a specific method used in algebra to factorize polynomials. It applies to polynomials that can be expressed in the form \(a^2 - b^2\). Such expressions are known as a 'difference' because of the subtraction and 'squares' because both terms are perfect squares. For example, \(x^2 - 9\) is a difference of squares because \(x^2\) and \(9\) (which is \(3^2\)) are perfect squares.

Formula: The equation is factored using the identity: \((a^2 - b^2) = (a + b)(a - b)\). This formula works because when the factored terms are expanded, they simplify back to the original polynomial:

For the given polynomial \(25d^2 - 64f^2\):

• Identify \(a = 5d\) and \(b = 8f\). Both are terms being squared.
• Factor it using the formula: \((5d + 8f)(5d - 8f)\).

Understanding this concept helps to recognize and factorize similar expressions easily.

polynomial factorization

Polynomial factorization involves breaking down a polynomial into simpler 'factor' polynomials that, when multiplied together, give the original polynomial. This process is crucial for solving polynomial equations, finding roots, and simplifying expressions.

In our example, \(25d^2 - 64f^2\), we see:

• Recognize the structure of the polynomial to apply an appropriate factoring method.
• Use the difference of squares method because the polynomial is in the form \(a^2 - b^2\). Here \(a\) is \(5d\) and \(b\) is \(8f\).
• Apply the factoring formula: \((a + b)(a - b)\). Hence, it factors into: \((5d + 8f)(5d - 8f)\).

Each polynomial factor is simpler, making the polynomial more manageable. This method is applied in various problems where polynomial simplification and solving are needed.

algebraic expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. Polynomials are a type of algebraic expression that include terms with variables raised to whole number exponents.

For instance, in \(25d^2 - 64f^2\), the terms \(25d^2\) and \(64f^2\) involve constants multiplied by variables raised to powers (squares in this case).

• Constants: Numbers without variables, like 25 and 64.
• Variables: Letters that represent numbers, such as \(d\) and \(f\).
• Exponents: The power to which a variable is raised. Here, both are raised to the power of 2.

Understanding how to manipulate these expressions by factoring and expanding them is crucial in solving algebraic equations. It’s the foundation for higher-level algebra concepts and many practical applications in science and engineering.