Problem 34 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Type of Polynomial

The given polynomial is a quadratic polynomial because it is in the form of a second-degree polynomial, written as \( ax^2 + bx + c \). Here, \( a = 1 \), \( b = -20 \), and \( c = 100 \).

Step 2: Check for Perfect Square Trinomial

A perfect square trinomial is a polynomial that can be expressed as \((x - h)^2 = x^2 - 2hx + h^2\).We check if \( (m - k)^2 = m^2 - 2km + k^2 \) fits the polynomial. Here, \( k = 10 \) because \( bisected\) or half of \( -20 \) is \( -10 \), and \( (-10)^2 = 100 \).

Step 3: Write the Factored Form

Since \( m^2 - 20m + 100 \) can be expressed as \((m - 10)^2\), this is a perfect square trinomial. The factored form is \((m - 10)(m - 10)\) or \((m-10)^2\).

Key Concepts

Quadratic Polynomials

A quadratic polynomial is a type of polynomial that includes variables raised to the second power. It follows the general expression, \( ax^2 + bx + c \) where \( a eq 0 \). This kind of polynomial is called "quadratic" because the highest degree a variable reaches is two. This polynomial can represent various situations ranging from physics problems to geometric shapes.
Commonly, solving quadratic polynomials may involve factoring, using the quadratic formula, or completing the square. Each method is valuable in finding the roots or solutions of the equation. When practicing with these polynomials, ensure to identify the coefficients \( a \), \( b \), and \( c \) properly, as they dictate which methods will likely simplify your problem-solving process.

Perfect Square Trinomials

Perfect square trinomials are special quadratic polynomials that can be written as the square of a binomial. This means that a polynomial of the form \( x^2 + 2hx + h^2 \) is a perfect square trinomial if it can be rewritten as \((x + h)^2\). Perfect square trinomials are useful because they simplify into binomials squared, which makes solving equations more straightforward.
To check if a polynomial is a perfect square trinomial:

• Identify \( b \) and see if it is twice the value of a proposed \( h \).
• Check if \( c \) equals the square of \( h \).

Applying these conditions to the polynomial \( m^2 - 20m + 100 \), we find that \( -20 = 2(-10) \) and \( 100 = (-10)^2 \). Thus, it can be expressed as \((m - 10)^2\), confirming it is a perfect square trinomial.

Factored Form

The factored form of a polynomial is a way of expressing it as a product of its factors. It provides insights into the roots or zeroes of the polynomial – the values at which the polynomial equals zero.
For quadratics, expressing them in factored form often simplifies them immensely, helping in solving quadratic equations. For a polynomial like \( m^2 - 20m + 100 \), identifying it as a perfect square trinomial allows us to write the factored form as \((m - 10)(m - 10)\) or \((m - 10)^2\).
This transformation from expanded form to factored form paves the way for solving equations more directly, either through setting each factor to zero or using them to analyze function behavior, such as graphing or optimization.

Second-Degree Polynomial

A second-degree polynomial is another term for a quadratic polynomial, as it involves terms where the highest power of the variable is two. These polynomials take the form \( ax^2 + bx + c \) and are foundational in algebra and calculus due to their wide application and straightforward properties, such as parabolic shapes.
Second-degree polynomials appear in various fields, modeling different behaviors or shapes, especially parabolic ones, in nature and technology. Recognizing the form allows mathematicians and scientists to apply various techniques to solve them, such as factoring or using graphing for visualization.
Overall, mastering second-degree polynomials involves understanding how they can convert between regular polynomial expressions and various factored or completed square forms to aid in deeper analysis or solutions.