Problem 34 - College Algebra

Question

For the following exercises, factor the polynomial. $$ m^{2}-20 m+100 $$

Step-by-Step Solution

Answer

The polynomial factors as \((m - 10)^2\).

Step 1: Identify the Quadratic Polynomial

The given polynomial is \(m^2 - 20m + 100\). This is a quadratic polynomial in standard form \(ax^2 + bx + c\), where \(a = 1\), \(b = -20\), and \(c = 100\).

Step 2: Check for Perfect Square Trinomial

To determine if the polynomial is a perfect square trinomial, check if \(b^2 = 4ac\). For this polynomial, \(b^2 = (-20)^2 = 400\) and \(4ac = 4 \cdot 1 \cdot 100 = 400\). Since \(b^2 = 4ac\), this polynomial is a perfect square trinomial.

Step 3: Factor the Polynomial

Since it is a perfect square trinomial, factor it as \((m - n)^2\), where \(n = \sqrt{c}\). Here, \(n = \sqrt{100} = 10\). Thus, the polynomial factors as \((m - 10)^2\).

Step 4: Verify the Factorization

Expand \((m - 10)^2\) to check if it equals the original polynomial. \((m - 10)(m - 10) = m^2 - 10m - 10m + 100 = m^2 - 20m + 100\). The factorization is correct.

Key Concepts

Quadratic Polynomial

A quadratic polynomial is any polynomial that can be expressed in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). In simple terms, it is a polynomial of degree 2, which means the highest exponent of the variable is 2. Quadratic polynomials often appear in problems involving parabolas, projectile motion, and optimization scenarios in middle and high school math.

• Standard form: The standard form of a quadratic polynomial is \(ax^2 + bx + c\).
• Leading coefficient: The coefficient \(a\) in the term \(ax^2\) is known as the leading coefficient.
• Importance: Quadratic polynomials are fundamental in mathematics due to their wide range of applications in real-life problems.

Understanding the structure of a quadratic polynomial is crucial as it lays the foundation for factoring and solving quadratic equations.

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic polynomial. It takes the form \((ax)^2 \pm 2abx + b^2\) and can be factored into \((ax \pm b)^2\). Recognizing a perfect square trinomial is a key skill in algebra because factoring these polynomials simplifies calculations.

• Form: The polynomial forms a perfect square trinomial if \(b^2 = 4ac\).
• Factorization: A perfect square trinomial like \(x^2 - 20x + 100\) can be factored into \((x - 10)^2\), since the middle term is twice the product of the terms \(x\) and \(b\).

Counterchecking your findings by squaring the factorized form can confirm your results and help ensure that the original polynomial was factored correctly.

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as the product of its factors. Factoring helps simplify polynomial expressions and solve polynomial equations. It is like breaking down a number into its prime factors, but in this case, it applies to polynomial expressions.

• Purpose: Simplifies expressions making them easier to work with.
• Methods: There are various methods like factoring by grouping, using the quadratic formula, or recognizing patterns such as perfect square trinomials.
• Verification: Always verify your factored form by multiplying the factors back together. This ensures the factorization is correct.

Mastering polynomial factorization requires practice and familiarity with different techniques, but once understood, it becomes an essential tool in solving algebraic problems.