Which Expression Is The Factorization Of X2 10x 21

Which Expression is the Factorization of x² + 10x + 21? A Comprehensive Guide

Factoring quadratic expressions is a fundamental skill in algebra. Understanding how to factor allows you to simplify expressions, solve equations, and delve deeper into more complex mathematical concepts. This comprehensive guide will explore the factorization of the quadratic expression x² + 10x + 21, providing a step-by-step explanation, exploring various methods, and addressing common pitfalls.

Understanding Quadratic Expressions

Before diving into the factorization, let's briefly review what a quadratic expression is. A quadratic expression is a polynomial of degree two, meaning the highest power of the variable (usually x) is 2. It generally takes the form ax² + bx + c, where a, b, and c are constants. In our case, a = 1, b = 10, and c = 21.

Method 1: Factoring by Inspection (Trial and Error)

This method involves finding two numbers that add up to the coefficient of the x term (b = 10) and multiply to the constant term (c = 21). Let's break it down:

Find the factors of 21: The pairs of factors of 21 are (1, 21), (3, 7), (-1, -21), and (-3, -7).
Identify the pair that adds up to 10: From the pairs above, only (3, 7) adds up to 10 (3 + 7 = 10).
Write the factored form: Since we found the numbers 3 and 7, the factored form of x² + 10x + 21 is (x + 3)(x + 7).

Verification: To verify this factorization, we can expand the expression (x + 3)(x + 7) using the FOIL method (First, Outer, Inner, Last):

First: x * x = x²
Outer: x * 7 = 7x
Inner: 3 * x = 3x
Last: 3 * 7 = 21

Combining these terms, we get x² + 7x + 3x + 21 = x² + 10x + 21. This confirms our factorization is correct.

Method 2: The AC Method (for more complex quadratics)

While the inspection method works well for simpler quadratics, the AC method is more systematic and can be applied to more complex expressions, even those where 'a' is not equal to 1.

Let's apply the AC method to x² + 10x + 21:

Identify a, b, and c: In our expression, a = 1, b = 10, and c = 21.
Calculate ac: ac = 1 * 21 = 21
Find two numbers that multiply to ac (21) and add up to b (10): As before, these numbers are 3 and 7.
Rewrite the expression: Rewrite the middle term (10x) as the sum of the two numbers found in step 3: x² + 3x + 7x + 21
Factor by grouping: Group the terms in pairs and factor out the greatest common factor (GCF) from each pair:

x(x + 3) + 7(x + 3)
Factor out the common binomial: Notice that (x + 3) is a common factor in both terms. Factor it out:

(x + 3)(x + 7)

This method confirms our previous factorization.

Method 3: Completing the Square (A More Advanced Technique)

Completing the square is a powerful technique used to solve quadratic equations and can also be applied to factorization. While it's more involved than the previous methods, understanding it provides a deeper insight into the nature of quadratic expressions.

Move the constant term to the right side: x² + 10x = -21
Take half of the coefficient of x, square it, and add it to both sides: Half of 10 is 5, and 5² = 25. Adding 25 to both sides:

x² + 10x + 25 = -21 + 25

x² + 10x + 25 = 4
Rewrite the left side as a perfect square trinomial: The left side is now a perfect square trinomial, which can be factored as (x + 5)².

(x + 5)² = 4
Take the square root of both sides:

x + 5 = ±√4 = ±2
Solve for x:

x = -5 ± 2

x = -3 or x = -7
Write the factored form: Since x = -3 and x = -7 are the roots, the factors are (x + 3) and (x + 7), leading to the factored form (x + 3)(x + 7).

Common Mistakes and How to Avoid Them

Several common mistakes can occur when factoring quadratic expressions. Let's address some of them:

Incorrectly identifying factors: Carefully consider all possible factor pairs of the constant term. It's easy to overlook a pair, particularly when dealing with larger numbers or negative factors.
Errors in the AC method: Ensure you correctly calculate 'ac' and find the appropriate pair of numbers that satisfy both the sum and product conditions. Double-checking your calculations is crucial.
Mistakes in completing the square: Accurately calculating half the coefficient of x and squaring it are essential steps. A small error here can significantly impact the result.
Forgetting to verify: Always verify your factorization by expanding the factored expression to ensure it matches the original quadratic expression.

Practical Applications of Factoring

Factoring quadratic expressions is not just a theoretical exercise; it has numerous practical applications across various fields:

Solving quadratic equations: Factoring is a key method for solving quadratic equations, enabling you to find the values of x that satisfy the equation.
Simplifying algebraic expressions: Factoring simplifies expressions, making them easier to understand and manipulate. This is particularly useful in calculus and other advanced mathematical subjects.
Graphing parabolas: The factored form of a quadratic expression reveals the x-intercepts (roots) of the corresponding parabola, which helps in accurately graphing the quadratic function.
Solving real-world problems: Quadratic equations model numerous real-world scenarios, such as projectile motion, area calculations, and optimization problems. Factoring allows for efficient solutions to these problems.

Conclusion

Factoring the quadratic expression x² + 10x + 21 results in (x + 3)(x + 7). This guide has explored three different methods – factoring by inspection, the AC method, and completing the square – each offering a unique approach to factorization. Mastering these methods is vital for success in algebra and its various applications. Remember to practice regularly, pay attention to detail, and always verify your results to build proficiency and confidence in factoring quadratic expressions. By understanding these techniques and avoiding common pitfalls, you can confidently tackle more complex quadratic expressions and apply this crucial skill to solve a wide array of mathematical problems.