Problem 41 - Beginning and Intermediate Algebra

Question

Factor by trial and error. $$21 d^{2}-22 d-8$$

Step-by-Step Solution

Answer

The factored form of the given quadratic expression, \(21 d^{2}-22 d-8\), is \((3d - 1)(7d + 8)\).

Step 1: Note the general form of the factored quadratic expression

We want to find two binomials in the form of (ad + e)(bd + f) such that $$(ad + e) (bd + f) = 21 d^{2} - 22 d - 8$$

Step 2: Factor the coefficients of d² and constant term

We will first find the factors of 21 (coefficient of \(d^2\)) and -8 (constant term). For 21, we have: 1 * 21 = 21 3 * 7 = 21 For -8, we have: -1 * 8 = -8 -2 * 4 = -8 1 * -8 = -8 2 * -4 = -8

Step 3: Experimenting with different combinations of factors

Now we will test different combinations of factors to find the two binomials that multiply to form the given quadratic expression. We will do this by matching the factors of 21 with factors of -8. Here are the possible combinations of factors: 1. (d + -1)(21d + 8) 2. (d + -2)(21d + 4) 3. (d + 1)(21d - 8) 4. (d + 2)(21d - 4) 5. (3d + -1)(7d + 8) 6. (3d + -2)(7d + 4) 7. (3d + 1)(7d - 8) 8. (3d + 2)(7d - 4) Let's test each combination to see if any of them satisfy the given quadratic expression: 1. (d - 1)(21d + 8) -> 21d² - 13d - 8 2. (d - 2)(21d + 4) -> 21d² - 38d - 8 3. (d + 1)(21d - 8) -> 21d² + 13d - 8 4. (d + 2)(21d - 4) -> 21d² + 38d - 8 5. (3d - 1)(7d + 8) -> 21d² - 22d - 8 6. (3d - 2)(7d + 4) -> 21d² - 20d - 8 7. (3d + 1)(7d - 8) -> 21d² + 20d - 8 8. (3d + 2)(7d - 4) -> 21d² + 22d - 8

Step 4: Match the correct combination of factors

From the testing in step 3, we can see that the correct combination is the fifth option: $$(3d - 1)(7d + 8) = 21d^2 - 22d - 8$$ So, the factored form of the given quadratic expression is: $$21 d^{2}-22 d-8= (3d - 1)(7d + 8)$$

Key Concepts

Trial and Error Method in Factoring

The trial and error method is an intuitive approach to factoring quadratic expressions. It's all about testing different combinations until you find the one that fits. This method can feel like solving a puzzle.

Here's how it works: When you have a quadratic expression like \(21d^2 - 22d - 8\), you want to break it into two binomials,

• Start by noting the coefficients of \(d^2\) and the constant term.
• Identify their possible factors.
• Test these factor combinations to see which ones work together.

The goal is to match these factor combinations to get back the original quadratic when multiplied.

While it can take several attempts to find the right pair, practicing this method helps you develop a better intuition for factoring over time.

Understanding Quadratic Expressions

Quadratic expressions are polynomials of degree 2, typically in the form \(ax^2 + bx + c\). In our example, the quadratic expression is \(21d^2 - 22d - 8\).

Key features of quadratics include:

• The highest power, or degree, is 2.
• The coefficients (like 21, -22) determine the expression's shape and roots.
• The constant term (-8) can affect the position in a graph.

Quadratic expressions appear in various applications, such as physics, engineering, and everyday problem-solving.

Understanding how to manipulate and factor them is vital for simplifying problems, finding solutions, and revealing more about their properties. Recognizing these elements can transform complex problems into more manageable forms.

Polynomial Factoring Techniques

Polynomial factoring is the process of breaking down polynomials into simpler components, or factors. This makes them easier to evaluate, solve, and understand.

For quadratics, like our expression \(21d^2 - 22d - 8\), the objective is to factor it into binomials:

• Consider potential factors of the coefficients.
• Use methods like trial and error to identify combinations that fit.

In this example, we identified the factors \((3d - 1)(7d + 8)\) by testing various combinations.

Factoring simplifies solving equations and algebraic manipulation and is a foundational skill in algebra and calculus. By mastering these techniques, complex expressions become more accessible and solvable.