Question

Factor each trinomial. State whether you used factoring by grouping or factoring by trial and error. (a) \(x^{2}+11 x+24\) (b) \(3 x^{2}+7 x-20\)

Short answer

The factored form of \(x^{2} + 11x + 24\) is \( (x + 8)(x + 3) \) using the trial and error method. The factored form of \(3x^{2} + 7x - 20\) is \( (3x - 5)(x + 4) \) using the method of factoring by grouping.

Step-by-step solution

Factoring \(x^{2} + 11x + 24\)

Look for two numbers that multiply to 24 (the constant term) and add to 11 (the coefficient of the middle term). The numbers are 8 and 3. Hence, the factored form of \(x^{2} + 11x + 24\) is \( (x + 8)(x + 3) \). Therefore, the method used here is trial and error since no grouping was necessary.

Factoring \(3x^{2} + 7x - 20\)

To factor \(3x^{2} + 7x - 20\), look for two numbers that multiply to -60 (which is 3 times -20) and add to 7. These numbers are 12 and -5. Recombine to form \(3x^{2} + 12x - 5x - 20\). Then, split into groups and factor by grouping, leading to \(3x(x + 4) - 5(x + 4)\). Hence, the factored form of \(3x^{2} + 7x - 20\) is \( (3x - 5)(x + 4) \).

Identification of the method used

The method used for \(x^{2} + 11x + 24\) was trial and error. The method used for \(3x^{2} + 7x - 20\) was factoring by grouping.

Key concepts

Factoring by Grouping

Factoring by grouping is an effective method for factoring polynomials that have four or more terms. This technique involves reorganizing and grouping the terms in order to find common factors in each group. Here's how it works:

1. **Arrange the terms:** Begin by ensuring you have a polynomial that can be successfully split into two groups.
2. **Create groups:** Separate the polynomial into two sets of terms. These groups should yield a common factor for each.
3. **Factor out the common factors:** Within each group, look for a factor that can be taken out of the group, simplifying the remaining expression.
4. **Find a common binomial:** Often, the terms remaining after factoring out in each group should now include a binomial common factor.
5. **Write the factored expression:** Combine these common binomials to display the expression fully factored.

For example, when factoring the trinomial \(3x^2 + 7x - 20\), after reorganizing, it becomes \(3x^2 + 12x - 5x - 20\). Grouping gives \(3x(x + 4) - 5(x + 4)\), where \((x + 4)\) is the common binomial factor, resulting in \((3x - 5)(x + 4)\). This method is particularly useful when trial and error does not easily yield a solution.

Factoring by Trial and Error

Factoring by trial and error is a simple and straightforward method often used for finding the factors of quadratic expressions like \(ax^2 + bx + c\) when these quadratics are easy to factor by intuition.

1. **Identify key coefficients:** Focus first on the end term \(c\) and the middle term \(b\) of the quadratic expression. You are looking for numbers that multiply to \(c\) and add to \(b\).
2. **Practice trial and error:** Experiment with different pairs of factors of \(c\) to find the combination that sums to \(b\).
3. **Write the expression:** Once you find such a pair, write your quadratic as a product of binomials.

In terms of \(x^2 + 11x + 24\), the task is simplified by identifying numbers that multiply to 24 and add to 11. The numbers 8 and 3 meet this criterion, allowing re-writing of the expression as \((x + 8)(x + 3)\). The simplicity of factors here made trial and error a practical choice.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, and they typically take the form \(ax^2 + bx + c\). These expressions are foundational components in algebra, representing parabolic graphs when plotted.

**Key Features of Quadratic Expressions:**
- **Degree:** The highest power of the variable, which is 2 in a quadratic expression.
- **Coefficients:** The constants \(a\), \(b\), and \(c\) each play unique roles in determining the shape and position of its graph.

**Factoring Quadratics:**
When tackling a quadratic expression, the goals often include determining its roots or zeros, which involve setting the expression equal to zero and factoring.

For example, when addressing \(3x^2 + 7x - 20\), this can be applied using various methods such as factoring by grouping or trial and error, depending on the simplicity of the expression. The roots or solutions are then the values of \(x\) that make the expression equal to zero, revealing the points where the parabola will intersect the x-axis.