Problem 61 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The polynomial factors to \((2x - 5)^2\).

Step 1: Identify the Quadratic Form

The polynomial given is \(4x^2 - 20x + 25\). We recognize that it follows the standard quadratic form \(ax^2 + bx + c\), where \(a = 4\), \(b = -20\), and \(c = 25\). Our task is to factor this expression if possible.

Step 2: Determine if it is a Perfect Square Trinomial

A perfect square trinomial is in the form \((px + q)^2 = p^2x^2 + 2pqx + q^2\). Here, we check if \(4x^2 - 20x + 25\) fits this form by expressing each term:- \(4x^2 = (2x)^2\)- \(25 = 5^2\)- Middle term \(-20x = 2 \cdot 2x \cdot 5\)This confirms that the polynomial can be written as the square of a binomial.

Step 3: Write the Polynomial as a Binomial Square

Given that \(4x^2 - 20x + 25\) fits the form \((px + q)^2\), we can conclude that the expression factors to \((2x - 5)^2\).

Step 4: Verify the Factored Form

To verify, expand \((2x - 5)^2\) back into the quadratic form:\[(2x - 5)(2x - 5) = 4x^2 - 10x - 10x + 25 = 4x^2 - 20x + 25\]Since the expanded form matches the original polynomial, our factorization is correct.

Key Concepts

Quadratic Form

Understanding the quadratic form is essential when dealing with polynomials. A quadratic polynomial typically has the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable. In the expression \(4x^2 - 20x + 25\), we identify that:

• \(a = 4\)
• \(b = -20\)
• \(c = 25\)

Recognizing the coefficients make it easier to apply various techniques in algebra, such as factoring or using the quadratic formula.
This polynomial represents a simple quadratic relation, which upon factorization can potentially transform into a product of linear factors. Identifying any hidden patterns like perfect squares empowers us to efficiently solve or simplify quadratic equations.

Perfect Square Trinomial

A perfect square trinomial is a specific type of quadratic form where the expression fits the format \((px + q)^2 = p^2x^2 + 2pqx + q^2\). This arises when two identical binomials are multiplied.

To verify if a quadratic is a perfect square trinomial, one should look for the following conditions:

• The first term is a perfect square, like \(4x^2 = (2x)^2\).
• The last term is a perfect square, such as \(25 = 5^2\).
• The middle term must match \(2pqx\), here \(-20x = 2 \cdot 2x \cdot 5\).

If these conditions are fulfilled, the quadratic can be expressed as \((2x - 5)^2\), confirming it is a perfect square trinomial.
Identifying this pattern simplifies the factorization of the polynomial by revealing its components right away.

Binomial Expansion

Binomial expansion involves expressing a binomial raised to a power as a multiple-term expression. For instance, expanding \((a + b)^2\) gives us \(a^2 + 2ab + b^2\). This method works by applying the distributive property systematically.

When factoring \(4x^2 - 20x + 25\), we use its identity as a perfect square and write it as \((2x - 5)^2\). To verify, we can expand using our knowledge of binomial expansion:

• \((2x - 5)(2x - 5)\) becomes \(4x^2 - 10x - 10x + 25\)
• We simplify the middle terms: \(-10x - 10x = -20x\)
• The result, \(4x^2 - 20x + 25\), matches the original polynomial

This illustrates how expanding can serve both as a solution and as a verification tool. Efficient use of binomial expansion confirms the polynomial factorization processes, showing both mastery and understanding of algebraic manipulations.