Problem 35 - College Algebra

Question

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

Step-by-Step Solution

Answer

The polynomial $m^2 - 20m + 100$ factors as $(m - 10)^2$.

Step 1: Recognize the Polynomial Form

The given polynomial is a quadratic polynomial in the standard form $ax^2 + bx + c$. Here, it is $m^2 - 20m + 100$, with $a=1$, $b=-20$, and $c=100$. First, recognize it as a perfect square trinomial.

Step 2: Identifying If It's a Perfect Square Trinomial

A perfect square trinomial takes the form \((x+d)^2 = x^2 + 2dx + d^2\). By comparing, think about what \(d\) would make the trinomial \(m^2 - 20m + 100\) a perfect square. Here \(d^2 = 100\), giving \(d = 10\), thus \(2d = 20\), matching \(-20\). That's \(m^2 - 2\times10\times m + 10^2.\)

Step 3: Factoring the Polynomial

Since we've confirmed that $m^2 - 20m + 100$ is a perfect square trinomial, we can factor it as $(m - 10)^2$. This uses the identity that $(x + d)^2 = x^2 + 2dx + d^2$.

Key Concepts

Quadratic Polynomial

Quadratic polynomials are expressions of degree 2, which can be identified through their standard form:

• \( ax^2 + bx + c \)

where:

• \( a, b, \) and \( c \) are constants,
• \( a \) is not equal to zero,
• \( x \) is the variable.

In our given exercise, \( m^2 - 20m + 100 \) is a quadratic polynomial with \( a = 1 \), \( b = -20 \), and \( c = 100 \). Recognizing a polynomial as quadratic is the foundational step in factoring or performing other algebraic operations. The presence of \( m^2 \) (a squared term) confirms it as a quadratic polynomial. Typically, these types of expressions are encountered in problems dealing with geometry, physics, and many types of applied mathematics.
Quadratic polynomials often form the basis for understanding more complex algebraic concepts.

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic polynomial where the expression can be expressed as the square of a binomial. To recognize a perfect square trinomial, you can look for a few key elements:

• The expression takes the form of \((x + d)^2 = x^2 + 2dx + d^2\).
• The first and last terms are perfect squares.
• The middle term is twice the product of the roots of the first and last terms.

In the given example, \( m^2 - 20m + 100 \), you can spot the pattern of a perfect square trinomial:

• \( m^2 \) is a perfect square (the square of \( m \)).
• \( 100 \) is a perfect square (the square of \( 10 \)).
• The middle term, \(-20m\), matches \( -2 \times 10 \times m \), fulfilling the formula's condition \( 2dx \).

Identifying perfect square trinomials simplifies both calculations and understanding of quadratic expressions. Once recognized, they can be factored with confidence, facilitating easier solution of quadratic equations.

Factoring Quadratic Equations

Factoring quadratic equations involves rewriting them as the product of simpler expressions. This process is particularly straightforward when dealing with perfect square trinomials.In the exercise, the equation \( m^2 - 20m + 100 \) was identified as a perfect square trinomial. Knowing this, you can factor it as:

• \((m - 10)^2\)

This result is achieved because \((m - 10)^2 = m^2 - 2 \times 10 \times m + 10^2\), perfectly matching the original polynomial. Factoring quadratic equations helps simplify expressions, solve equations algebraically, and better understand their roots and graphical representations.
Learning this skill is valuable in numerous mathematical applications, enabling the breakdown and analysis of more complex equations.