Problem 20 - Algebra 2

Question

Factor each polynomial. \(5 x^{2}-80\)

Step-by-Step Solution

Answer

The polynomial \(5x^2 - 80\) factors to \(\sqrt{5}(x - 4)(x + 4)\).

Step 1: Identify the Type of Polynomial

The polynomial given is a quadratic polynomial, as it is of the form \(ax^2 + bx + c\). In this particular case, it is a difference of squares because it can be rewritten with perfect square terms.

Step 2: Rewrite as a Difference of Squares

Observe that the expression can be written in the form \(a^2 - b^2\). In this case, \(5x^2 - 80\) is equivalent to \([\sqrt{5}x]^2 - [\sqrt{80}]^2\). Simplify \(\sqrt{80}\) to \(4\sqrt{5}\). Thus, we rewrite it as \((\sqrt{5}x)^2 - (4\sqrt{5})^2\).

Step 3: Factor the Difference of Squares

Use the identity \(a^2 - b^2 = (a - b)(a + b)\). Let \(a = \sqrt{5}x\) and \(b = 4\sqrt{5}\), so, \((\sqrt{5}x - 4\sqrt{5})(\sqrt{5}x + 4\sqrt{5})\).

Step 4: Simplify the Expression

Notice the common factor of \(\sqrt{5}\) in both expressions to see if further simplification is possible. Rewrite it as \(\sqrt{5}(x - 4)(x + 4)\) after factoring \(\sqrt{5}\) out.

Key Concepts

Difference of Squares

In algebra, the "Difference of Squares" is a specific kind of expression where two square terms are subtracted. It takes the form \(a^2 - b^2\), and this identity is incredibly useful for factoring. This can be broken down into two binomials: \((a - b)(a + b)\).

The essence of the difference of squares lies in its simplicity: it reduces complex algebraic expressions into manageable pieces. For instance, the expression \(x^2 - 9\) is easily factored as \((x - 3)(x + 3)\).

To apply the difference of squares to a polynomial:

• Ensure both terms are perfect squares.
• Subtract the square terms.
• Use \(a^2 - b^2 = (a - b)(a + b)\) to factorize.

Identifying and utilizing this pattern streamlines the factoring process, saving time and simplifying calculations.

Quadratic Polynomial

A "Quadratic Polynomial" is an algebraic expression of degree 2, typically in the form \(ax^2 + bx + c\). It is the highest power of the variable that determines the name "quadratic," derived from "quadratus," meaning square.

Quadratic polynomials can appear in three standard types:

• Standard form: \(ax^2 + bx + c\)
• Factored form: \((px + q)(rx + s)\)
• Vertex form: \(a(x - h)^2 + k\)

Each provides different insights into the polynomial's behavior and roots.

The quadratic polynomial \(5x^2 - 80\) identifies as a difference of squares once transformed into its factorable form. Recognizing such patterns can aid in rapidly solving problems related to quadratic expressions in various mathematical contexts.

Algebraic Expressions

"Algebraic Expressions" consist of variables, numbers, and operations. They form the building blocks of algebra and are the means by which mathematical relationships and properties are conveyed. For example, \(5x^2 - 80\) includes constants (5 and 80), a variable (x), and mathematical operations (subtraction).

Key components of an algebraic expression include:

• Terms: The individual parts separated by plus or minus signs.
• Coefficients: Numerical factors of terms involving variables.
• Variables: Symbols representing unknown quantities.

Algebraic expressions can often be simplified or factored to reveal underlying structures and relationships, enabling a deeper understanding, as seen in the example transforming \(5x^2 - 80\) into \((\sqrt{5}(x - 4)(x + 4))\). Understanding these components aids in manipulating and simplifying any algebraic problem.

Polynomial Identities

"Polynomial Identities" are equations that hold true for all values of the variables within certain constraints. These identities simplify polynomial expressions and help with solving equations. Common identities include the difference of squares, perfect square trinomials, and sum or difference of cubes.

For example, the identity \(a^2 - b^2 = (a - b)(a + b)\) demonstrates a fundamental strategy for algebraic manipulation: transforming complex expressions into more manageable forms.

To leverage polynomial identities effectively:

• Identify the pattern within the expression.
• Apply the corresponding identity.
• Simplify to achieve the desired form.

Recognizing and applying these identities allow for efficient problem-solving in algebraic contexts, as seen in the factorization of \(5x^2 - 80\) using the difference of squares identity. Mastery of these identities is key to navigating higher-level mathematics.