Which Expression Is A Factor Of 12x2 29x 8

Which Expression is a Factor of 12x² + 29x + 8? A Comprehensive Guide

Factoring quadratic expressions is a fundamental skill in algebra. Understanding how to factor allows you to simplify equations, solve for unknowns, and delve deeper into mathematical relationships. This article will provide a thorough explanation of how to determine which expression is a factor of 12x² + 29x + 8, covering various factoring techniques and offering insightful strategies for similar problems.

Understanding Quadratic Expressions

Before diving into the factorization, let's solidify our understanding of quadratic expressions. 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, we have the quadratic expression 12x² + 29x + 8, where a = 12, b = 29, and c = 8.

Methods for Factoring Quadratic Expressions

Several methods exist for factoring quadratic expressions. The most common are:

1. The AC Method (also known as the grouping method)

This is a widely used method, especially for quadratic expressions where the coefficient of x² (a) is not equal to 1. The steps are as follows:

Find the product 'ac': In our example, ac = 12 * 8 = 96.
Find two numbers that add up to 'b' and multiply to 'ac': We need two numbers that add up to 29 and multiply to 96. These numbers are 3 and 32 (3 + 32 = 35, there's a mistake here, let's try again. The correct pair is 3 and 32, but we made a mistake in our previous calculation. It should be 16 and 3 because 16 + 3 =19, not 29. Let's adjust the steps to reflect this.). After careful consideration, we find that 16 and 3 are the required numbers (16 + 3 = 19, not 29, so this method might not be ideal for this specific case). Let's move on to the next method.

2. Trial and Error Method

This method involves systematically trying different combinations of factors until we find the correct one. Since we have 12x² + 29x + 8, we need to find two binomials that, when multiplied, result in this expression.

We consider the factors of 12 (1, 12; 2, 6; 3, 4) and the factors of 8 (1, 8; 2, 4). Let's try various combinations:

(3x + 1)(4x + 8): This expands to 12x² + 24x + 4x + 8 = 12x² + 28x + 8 (Incorrect)
(3x + 8)(4x + 1): This expands to 12x² + 3x + 32x + 8 = 12x² + 35x + 8 (Incorrect)
(3x + 2)(4x + 4): This expands to 12x² + 12x + 8x + 8 = 12x² + 20x + 8 (Incorrect)
(6x + 1)(2x + 8): This expands to 12x² + 48x + 2x + 8 = 12x² + 50x + 8 (Incorrect)
(6x + 8)(2x + 1): This expands to 12x² + 6x + 16x + 8 = 12x² + 22x + 8 (Incorrect)
(1x + 8)(12x + 1): This expands to 12x² + x + 96x + 8 = 12x² + 97x + 8 (Incorrect)
(x+1)(12x+8): This expands to 12x² + 8x + 12x + 8 = 12x²+20x+8
(x+2)(12x+4): This expands to 12x² + 4x + 24x + 8 = 12x²+28x+8
(3x + 8)(4x + 1): This expands to 12x² + 3x + 32x + 8 = 12x² + 35x + 8 (Incorrect)
(3x + 1)(4x + 8): This expands to 12x² + 24x + 4x + 8 = 12x² + 28x + 8 (Incorrect)
(x+8)(12x+1): This expands to 12x² + 97x + 8 (Incorrect)

After exploring various combinations, we find that none of the simple factor combinations yield the original quadratic. It's possible the expression is not factorable using simple integer coefficients. We might need to use the quadratic formula to find the roots and then work backward to obtain the factors.

Let's consider the possibility that there's a typo in the original expression. A small change could make it readily factorable.

Let's assume, for the sake of illustration, that the expression was meant to be 12x² + 35x + 8. This is indeed factorable:

(3x + 8)(4x + 1) = 12x² + 3x + 32x + 8 = 12x² + 35x + 8

Therefore, in this corrected example, (3x + 8) and (4x + 1) are factors of 12x² + 35x + 8.

Using the Quadratic Formula

If the trial and error method proves unsuccessful, the quadratic formula is a reliable method to find the roots of the quadratic equation ax² + bx + c = 0. The roots are given by:

x = [-b ± √(b² - 4ac)] / 2a

For our original expression 12x² + 29x + 8 = 0:

x = [-29 ± √(29² - 4 * 12 * 8)] / (2 * 12) x = [-29 ± √(841 - 384)] / 24 x = [-29 ± √457] / 24

These are irrational roots, indicating that the original expression likely doesn't factor nicely using integers.

Conclusion

Determining factors for quadratic expressions involves systematic application of factoring methods. The AC method and trial and error are common approaches. However, some quadratic expressions, like the original 12x² + 29x + 8, might not be easily factorable using simple integer coefficients. The quadratic formula provides a foolproof method to find roots, which can then be used to express the quadratic in factored form, although the factors may involve irrational numbers. Always double-check the original expression for potential errors; a minor typographical mistake can significantly affect the factorization process. This thorough examination illustrates the importance of utilizing multiple approaches and understanding the implications of irrational roots in quadratic factorization.