Problem 17 - Precalculus: Functions and Graphs

Question

Factor the polynomial. $$4 x^{2}-20 x+25$$

Step-by-Step Solution

Answer

The polynomial can be factored as \((2x - 5)^2\).

Step 1: Understand the Problem

We need to factor the quadratic polynomial \(4x^2 - 20x + 25\). Factoring means expressing the polynomial as a product of simpler polynomials, usually linear factors.

Step 2: Check for a Perfect Square

Observe the form of the quadratic: \(4x^2 - 20x + 25\). Notice the coefficients 4 and 25 are perfect squares. This suggests a potential perfect square trinomial form as \((ax + b)^2\).

Step 3: Identify the Perfect Square Components

The polynomial \(4x^2 - 20x + 25\) could be written as \((2x)^2 - 2 \cdot 2x \cdot 5 + 5^2\), where \(a = 2x\) and \(b = 5\). This matches the pattern \((a - b)^2 = a^2 - 2ab + b^2\).

Step 4: Factor the Polynomial

Using the perfect square trinomial formula, the polynomial can be factored as \((2x - 5)^2\).

Step 5: Verification

Verify the factorization by expanding \((2x - 5)(2x - 5)\) back to \(4x^2 - 20x + 25\) to ensure it matches the original polynomial.

Key Concepts

Quadratic Polynomial

A quadratic polynomial is a polynomial of degree two. This means it includes a term with the variable raised to the second power, such as in the expression \(ax^2 + bx + c\). Quadratic polynomials are among the simplest type of polynomial equations and can describe parabolic graphs on a coordinate plane.

• The general form is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).
• The "\(ax^2\)" term is what gives the polynomial its quadratic nature.
• In our exercise, the polynomial \(4x^2 - 20x + 25\) fits this format, making it a quadratic polynomial, with \(a = 4\), \(b = -20\), and \(c = 25\).

Understanding quadratic polynomials is crucial for solving various mathematical problems, as they appear frequently in areas like physics, economics, and engineering.

Perfect Square Trinomial

A perfect square trinomial is a special form of quadratic polynomial that can be expressed as the square of a binomial. Recognizing a perfect square trinomial makes factoring much simpler because it follows a predictable pattern.

• The pattern for a perfect square trinomial is \((ax + b)^2 = a^2x^2 + 2abx + b^2\).
• In the given exercise \(4x^2 - 20x + 25\), identifying if it is a perfect square involves checking if it fits the pattern \(a^2x^2 - 2abx + b^2\). Here \(a = 2\) and \(b = 5\) because \(4 = (2)^2\) and \(25 = (5)^2\).
• The middle term \(-20x\) matches \(-2abx\), since \(-2 \times 2 \times 5 = -20\).

This detection makes factoring straightforward by acknowledging it as the square of \((2x - 5)\).

Factoring Techniques

Factoring techniques are strategies used to express polynomials as a product of simpler polynomials. This often involves recognizing common patterns or applying specific formulas.

• The simplest form of factoring a quadratic is looking for two numbers that multiply to \(a \times c\) and add to \(b\). This is especially useful for simple quadratics.
• For recognizing perfect square trinomials, confirm that the first and last terms are perfect squares and that the middle term equals \(2 \times \) (the square root of the first term) \(\times \) (the square root of the last term).
• Our example polynomial fits the pattern \((ax - b)^2\), simplifying our work to just rewriting it as \((2x - 5)^2\).

Mastering these techniques enables efficient solving of polynomial equations and forms a foundation for more advanced algebra topics.

Mathematical Verification

Mathematical verification involves confirming that a factorized expression indeed represents the original polynomial. This step ensures that no errors were made during the factoring process.

• After factoring \(4x^2 - 20x + 25\) as \((2x - 5)^2\), we need to verify by expanding it back.
• Expanding \((2x - 5)(2x - 5)\) gives \(4x^2 - 20x + 25\).
• Each expanded term should match the respective terms of the original polynomial: \(4x^2\), \(-20x\), and \(25\).

By checking the expanded result against the original quadratic polynomial, we confirm the accuracy of the factorization. Verification is crucial as it provides validity to the solution, ensuring the factoring was correctly executed.