Problem 34 - Algebra A Combined Function

Question

Factor each trinomial completely. Some of these trinomials contain a greatest common factor (other than 1). Don't forget to factor out the GCF first. $$ r^{2}-10 r+21 $$

Step-by-Step Solution

Answer

The trinomial factors as \((r - 7)(r - 3)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, look for the greatest common factor (GCF) of all the terms in the trinomial. The given trinomial is \(r^2 - 10r + 21\). Each coefficient is 1, -10, and 21. The GCF of these coefficients and the variables is 1, implying there is no common factor to factor out first. Therefore, we can proceed without factoring out any GCF.

Step 2: Factor the Trinomial

Next, we need to factor the trinomial \(r^2 - 10r + 21\). Look for two numbers that multiply to the constant term (21) and add to the linear coefficient, which is -10. The numbers -7 and -3 multiply to \(21\) and add up to \(-10\). Thus, the trinomial factors as \((r - 7)(r - 3)\).

Step 3: Write the Final Factored Form

Combine the factors identified in the previous step to write the completely factored form of the trinomial. The factored form of \(r^2 - 10r + 21\) is \((r - 7)(r - 3)\).

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is an important concept in factoring polynomials, as it refers to the largest number or term that can divide each coefficient or term in an expression without leaving a remainder. When factoring trinomials, the first step is often to identify any GCFs among the terms. In this case, considering the expression \(r^2 - 10r + 21\), we examine the coefficients 1, -10, and 21. The GCF of these numbers is 1 since no larger integer divides them. Although in this instance, no GCF greater than 1 exists, always check for a GCF first in order to simplify the expression where possible. Identifying and factoring out the GCF can make subsequent steps easier, potentially reducing complex expressions into simpler forms. This can sometimes reveal further factorizations that aren't initially obvious.

Quadratic Expressions

Quadratic expressions are polynomials of degree 2, generally represented as \(ax^2 + bx + c\). In these expressions, "a" is the coefficient of the squared term, "b" is the coefficient of the linear term, and "c" is the constant term. In the exercise given, the quadratic expression is \(r^2 - 10r + 21\), where the coefficient of the squared term "r" (a) is 1, "b" is -10, and "c" is 21. Factoring quadratic expressions involves rewriting them as the product of two binomials. The goal is to find two numbers whose product equals the constant term (c) and whose sum equals the linear coefficient (b). This process is crucial for simplifying expressions and solving quadratic equations. In our example, we found that the numbers needed are -7 and -3, since \((-7) + (-3) = -10\) and \((-7) \times (-3) = 21\). Thus, \(r^2 - 10r + 21\) factors into \((r - 7)(r - 3)\).

Completing the Square

Completing the square is an alternative method to factor quadratic expressions. This technique involves rearranging the expression to form a perfect square trinomial, which can then be expressed as a binomial squared.To complete the square, first ensure the quadratic expression is in the form \(ax^2 + bx + c\) and check if a is 1, as in our example \(r^2 - 10r + 21\). Then, consider only the quadratic and linear terms (ignoring "c") and create a new term by taking half of the linear coefficient (b), squaring it, and adjusting for it.For \(r^2 - 10r+ 21\), take half of -10 to get -5, then square it to get 25. Add and subtract 25 inside the expression: \[r^2 - 10r + 25 - 25 + 21\] Reorganizing gives: \[(r - 5)^2 - 4\] While completing the square isn't necessary for this factorization, it's a useful tool for solving quadratic equations, especially when they don't factor neatly into integers. This method also sets the stage for understanding the derivation of the quadratic formula.