Question

Factor the polynomial completely. $$ 49 k^4-9 $$

Short answer

The factored form of the polynomial \(49k^4 - 9\) is \((\sqrt{7}k - \sqrt{3})^2(\sqrt{7}k + \sqrt{3})^2\).

Step-by-step solution

Identify the squares

Rewrite the given polynomial as \((7k^2)^2 - 3^2\). This clearly shows the difference of two squares.

Apply the formula for difference of squares first time

Using the difference of squares formula \(a^2 - b^2 = (a - b)(a + b)\), rewrite the equation as \((7k^2 - 3)(7k^2 + 3)\).

Apply the formula for difference of squares second time

Observe that both \(7k^2 - 3\) and \(7k^2 + 3\) are the difference and sum of squares respectively. Apply the formula to these terms separately to obtain \((\sqrt{7}k - \sqrt{3})(\sqrt{7}k + \sqrt{3})(\sqrt{7}k - \sqrt{3})(\sqrt{7}k + \sqrt{3})\). Note that \(\sqrt{7}k\) is the square root of \(7k^2\) and \(\sqrt{3}\) is the square root of 3.

Simplify the expression

From step 3, note that \((\sqrt{7}k - \sqrt{3})(\sqrt{7}k + \sqrt{3})\) and \((\sqrt{7}k - \sqrt{3})(\sqrt{7}k + \sqrt{3})\) are identical. Therefore, the expression can be written as \((\sqrt{7}k - \sqrt{3})^2(\sqrt{7}k + \sqrt{3})^2\).

Key concepts

Difference of Squares

The term "difference of squares" refers to a specific pattern in algebraic expressions. This pattern appears as: \( a^2 - b^2 \), where both \( a \) and \( b \) have been squared. The beauty of this pattern lies in its ability to simplify into a much more manageable form. It can be factored into \((a - b)(a + b)\). This transformation holds great significance in algebra as it resolves what could be complex quadratic formulations into simple linear factors.
Identifying the difference of squares requires recognizing terms that are perfect squares. For example, in the given polynomial \( 49k^4 - 9 \), both \( 49k^4 \) and \( 9 \) are perfect squares. We express them as \((7k^2)^2\) and \((3)^2\) respectively, clearly indicating the difference of squares pattern. Therefore, we apply the formula as shown in the step-by-step solution to simplify the polynomial.

Polynomial Expressions

Polynomial expressions are algebraic expressions made up of variables and coefficients, arranged in terms of non-negative integer powers. They can have one or many terms. The expression in the exercise, \( 49k^4 - 9 \), is a binomial with two terms: \( 49k^4 \) and \(-9\). Each of these terms is part of a polynomial structure.
When dealing with polynomial expressions, recognizing their form is crucial. It often involves looking for common patterns such as the difference of squares, which can make factorization easier. Recognizing these patterns helps in breaking down complex expressions into simpler components. Understanding polynomial structures, and manipulating them algebraically, is a fundamental skill in algebra and serves as the foundation for solving more complex equations.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying algebraic expressions using various algebraic techniques. Techniques include factoring, expanding, or using formulas like the one for the difference of squares. It's a core part of solving algebra problems and helps transform difficult equations into simpler, solvable forms.
In the provided exercise, algebraic manipulation is used extensively. First, identifying the difference of squares enables us to factor the polynomial expression into simpler terms. With the polynomial \( 49k^4 - 9 \), we applied the difference of squares formula: \((7k^2 - 3)(7k^2 + 3)\), which is further broken down using the square root operations, demonstrating powerful manipulation skills.

• Transforms complex polynomials into manageable factors.
• Leverages known formulas for simplification.

Effective algebraic manipulation not only aids factorization but also helps in solving equations, deriving functions, and even in calculus. Understanding these concepts equips students with the tools necessary to tackle diverse mathematical challenges.