Problem 31 - College Algebra and Calculus: An Applied Approach

Question

Factor the trinomial.\(x^{2}-30 x+200\)

Step-by-Step Solution

Answer

The factored form of the trinomial \(x^2 - 30x + 200\) is \((x-20)(x-10)\).

Step 1: Identify the Coefficients

Identify the coefficients and the constant term in the given trinomial. Here, the coefficient of \(x^2\) (a) is 1, the coefficient of x (b) is -30 and the constant term (c) is 200.

Step 2: Find the numbers

Find two numbers that multiply to 200 and add to -30. In this case, these numbers are -20 and -10 (-20 * -10 = 200 and -20 + (-10) = -30).

Step 3: Factor the trinomial

Express the trinomial in factored form. Replace \(x^2 - 30x + 200\) with \((x-20)(x-10)\). These are the two binomial factors of the trinomial.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental concept in algebra. They are polynomials of degree two, characterized by the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). Solving quadratic equations can range from simple factoring to using the quadratic formula. Notably, the solutions of quadratic equations are known as "roots." These roots can be real or complex numbers, depending on the discriminant \( b^2 - 4ac \). When \( b^2 - 4ac > 0 \), the equation has two distinct real roots. If \( b^2 - 4ac = 0 \), there is exactly one real root (a repeated root), and when \( b^2 - 4ac < 0 \), the equation has two complex roots. By understanding this, students can predict the number and type of solutions their quadratic will have. It's essential to bear these concepts in mind particularly when engaging with trinomials, as they frequently appear in tasks associated with quadratic equations.

Algebraic Expressions

Algebraic expressions are composed of variables, constants, and operations such as addition, subtraction, multiplication, and division. The expression \( x^2 - 30x + 200 \) is an example of a quadratic expression. It consists of a squared term \( x^2 \), a linear term \( -30x \), and a constant term \( 200 \). Understanding how to manipulate these expressions is crucial in algebra. Simplifying them often involves combining like terms or factoring to make expressions more workable. Factoring algebraic expressions can reveal important characteristics of a mathematical problem. For instance, when you factor \( x^2 - 30x + 200 \) into \((x-20)(x-10)\), it unveils the roots of the associated quadratic equation.

Binomial Factors

Binomial factors are expressions containing two terms and play a significant role while factoring trinomials. The goal of factoring trinomials is to express them as a product of binomials, as seen with \( x^2 - 30x + 200 \), which factors into \((x-20)(x-10)\). The factors \((x-20)\) and \((x-10)\) are each binomials consisting of two distinct terms.
When factoring, identifying suitable binomial factors requires finding two numbers that multiply to give the constant term and add up to give the linear coefficient.

• Multiplication must yield the trinomial's constant term; here, \( -20 \times -10 = 200 \).
• Addition must equal the trinomial's linear coefficient; in this example, \( -20 + (-10) = -30 \).

This method simplifies the trinomial into a more manageable form, illustrating how the expression is structured mathematically. Understanding binomial factors is therefore vital for mastering the process of factoring trinomials.