Problem 37 - 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. $$ 3 x^{2}+9 x-30 $$

Step-by-Step Solution

Answer

The trinomial factors to \(3(x + 5)(x - 2)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, examine the coefficients of the trinomial terms: 3, 9, and 30. The greatest common factor among them is 3. Therefore, the GCF of the trinomial is 3.

Step 2: Factor Out the GCF

Divide each term of the trinomial by the GCF (3). This gives: \[ 3(x^2 + 3x - 10) \] Now, work with the expression inside the parentheses: \(x^2 + 3x - 10\).

Step 3: Factor the Quadratic Expression

To factor \(x^2 + 3x - 10\), look for two numbers that multiply to \(-10\) and add to \(3\). These numbers are 5 and -2. Thus, you can express the trinomial as:\[ x^2 + 3x - 10 = (x + 5)(x - 2) \]

Step 4: Write the Complete Factored Form

Combine the factored quadratic expression with the GCF. The completely factored form of the original trinomial is:\[ 3(x + 5)(x - 2) \]

Key Concepts

Greatest Common Factor

A greatest common factor (GCF) in algebra is the highest number that divides exactly into each of the numbers in a set. For trinomials, we focus on the coefficients of each term. Calculating the GCF is the first step in factoring algebraic expressions, such as trinomials, to simplify the equation and make further factoring easier and possible. To find the GCF:

• List the factors of each coefficient in the expression. For example, with 3, 9, and 30, the factors are 3, 9, and 30 themselves.
• Identify the largest factor that appears in each list. In our example, the largest common factor is 3.

Once the GCF is determined, it is factored out of the trinomial, dividing each term by this factor. This makes the remaining expression simpler to factor further.

Quadratic Expressions

Quadratic expressions are algebraic expressions where the highest power of the variable is squared (to the power of 2). They take the general form of:\[ ax^2 + bx + c \]where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable.Factoring quadratic expressions involves finding two binomials that multiply to form the quadratic. To factor, we look for two numbers whose product equals the constant term \(c\), and whose sum equals \(b\), the coefficient of the linear term \(x\).In our case:

• We need two numbers that multiply to give \(-10\) (given the quadratic expression \(x^2 + 3x - 10\)) and add to \(3\).
• Here, \(5\) and \(-2\) fit both conditions: \(5 \times -2 = -10\) and \(5 + (-2) = 3\).

Thus, the quadratic is factored into \((x + 5)(x - 2)\), making it easier to handle algebraically.

Algebraic Expressions

Algebraic expressions combine variables, coefficients, and arithmetic operations into a single expression. These can include polynomials like trinomials, where the expression is composed of three terms. Factoring algebraic expressions, particularly trinomials, is a helpful method in algebra that simplifies expressions for increased ease in solving equations. When working with trinomials:

• First check for a greatest common factor (GCF) to simplify calculations. This initial step reduces the complexity of the expression.
• Next, factor any quadratic expressions involved, using methods such as the one previously described to split the quadratic into two binomials.

Successfully factoring the trinomial allows for solving equations, simplifying algebraic fractions, and finding roots more conveniently. Understanding the structure and form of algebraic expressions is crucial in mastering basic algebraic manipulation.