Illegal Values Of C In A Multiplication Statement

by Alex Johnson 50 views

In mathematics, identifying illegal values within expressions is crucial, especially when dealing with fractions. These illegal values often lead to undefined expressions, such as division by zero. In this article, we will explore how to pinpoint these values in a given multiplication statement. Our focus will be on understanding the underlying principles and applying them to solve the problem effectively. Let's dive into the concept of illegal values and how to find them, ensuring a solid grasp of this essential mathematical skill.

Understanding Illegal Values

In the context of rational expressions, illegal values are those that make the denominator of a fraction equal to zero. This is because division by zero is undefined in mathematics. Identifying these values is crucial for several reasons. Firstly, it ensures that we're working within the bounds of mathematical rules, preventing errors in our calculations. Secondly, it helps us understand the domain of the function represented by the expression, which is the set of all possible input values for which the function produces a valid output. When we encounter an expression like the one in the question, the first step is to factorize the polynomials in both the numerator and the denominator. This allows us to see the structure of the expression more clearly and identify potential values of c that might make the denominator zero. Remember, a polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. Factoring a polynomial means breaking it down into simpler expressions that, when multiplied together, give the original polynomial. This process is essential for simplifying rational expressions and solving equations involving polynomials. So, before we can determine the illegal values, we need to master the art of factoring.

The Multiplication Statement

Let's consider the multiplication statement given:

c2−3c−10c2+5c−14⋅c2−c−2c2−2c−15\frac{c^2-3 c-10}{c^2+5 c-14} \cdot \frac{c^2-c-2}{c^2-2 c-15}

To find the illegal values of c, we need to identify the values that make any of the denominators equal to zero. The denominators in this expression are c2+5c−14c^2 + 5c - 14 and c2−2c−15c^2 - 2c - 15. Therefore, we need to solve the equations c2+5c−14=0c^2 + 5c - 14 = 0 and c2−2c−15=0c^2 - 2c - 15 = 0. Solving these quadratic equations will give us the values of c that we must exclude from the domain of the expression. Remember, the domain of a function is the set of all possible input values (in this case, c values) for which the function produces a valid output. By identifying the illegal values, we are essentially defining the boundaries of this domain. Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable is 2. They have a general form of ax2+bx+c=0ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable. There are several methods for solving quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, factoring is a straightforward approach, but for more complex equations, the quadratic formula might be necessary.

Factoring the Denominators

Factoring c2+5c−14c^2 + 5c - 14

We need to find two numbers that multiply to -14 and add to 5. Those numbers are 7 and -2. So, we can factor the quadratic as follows:

c2+5c−14=(c+7)(c−2)c^2 + 5c - 14 = (c + 7)(c - 2)

Setting this equal to zero gives us:

(c+7)(c−2)=0(c + 7)(c - 2) = 0

This implies that c=−7c = -7 or c=2c = 2 are illegal values because they make the denominator zero.

Factoring a quadratic expression involves breaking it down into two binomial factors. This is the reverse process of expanding two binomials using the distributive property (often remembered by the acronym FOIL - First, Outer, Inner, Last). The key to factoring is finding the correct pair of numbers that satisfy the conditions related to the coefficients of the quadratic. In this case, we needed two numbers that multiply to the constant term (-14) and add up to the coefficient of the linear term (5). Once we find these numbers, we can easily construct the factored form of the quadratic. This skill is fundamental in algebra and is used extensively in solving equations and simplifying expressions.

Factoring c2−2c−15c^2 - 2c - 15

Similarly, we need two numbers that multiply to -15 and add to -2. Those numbers are -5 and 3. Thus, the factored form is:

c2−2c−15=(c−5)(c+3)c^2 - 2c - 15 = (c - 5)(c + 3)

Setting this equal to zero gives us:

(c−5)(c+3)=0(c - 5)(c + 3) = 0

This means c=5c = 5 or c=−3c = -3 are also illegal values.

The process of factoring c2−2c−15c^2 - 2c - 15 is analogous to the previous example. We again look for two numbers that multiply to the constant term (-15) and add up to the coefficient of the linear term (-2). Finding these numbers requires a bit of trial and error, but with practice, it becomes more intuitive. Once we have the correct pair (-5 and 3), we can write the factored form as (c−5)(c+3)(c - 5)(c + 3). Setting each factor to zero allows us to identify the values of c that make the denominator zero, which are the illegal values we are seeking. This consistent approach to factoring quadratic expressions is a cornerstone of algebraic manipulation.

Identifying Illegal Values

From the factored denominators, we found the following illegal values:

  • c=−7c = -7
  • c=2c = 2
  • c=5c = 5
  • c=−3c = -3

Therefore, the illegal values of c in the given multiplication statement are -7, -3, 2, and 5.

To summarize, the illegal values are those that make the denominator of any fraction in the expression equal to zero. These values are excluded from the domain of the expression because division by zero is undefined. We identified these values by factoring the denominators, setting each factor equal to zero, and solving for c. This process ensures that we have a clear understanding of the values that cannot be used in the expression. This step is crucial in various mathematical contexts, including solving equations, graphing functions, and simplifying expressions. By identifying and excluding these values, we ensure that our mathematical operations are valid and our results are accurate.

The Correct Answer

Comparing our results with the given options, we see that the correct answer is:

  • D) c=−7,c=−3,c=2c = -7, c = -3, c = 2, and c=5c = 5

This option accurately lists all the illegal values of c that we identified through our factoring and equation-solving process. It's essential to verify our solution against the given options to ensure that we have correctly identified all the illegal values and have not missed any. This step also serves as a check on our calculations and reasoning, ensuring that our final answer is accurate and aligns with the problem's requirements. The ability to correctly identify and apply the appropriate mathematical techniques, such as factoring and solving equations, is crucial for success in algebra and beyond.

Conclusion

Finding illegal values in rational expressions is a fundamental skill in algebra. It ensures that we avoid division by zero and work within the valid domain of the expression. By factoring the denominators and solving for the values that make them zero, we can effectively identify these illegal values. This process is essential for simplifying expressions, solving equations, and understanding the behavior of functions. Remember to always check for illegal values when dealing with rational expressions to ensure the accuracy and validity of your mathematical work.

To further enhance your understanding of rational expressions and illegal values, explore resources like Khan Academy's article on Rational Expressions for comprehensive lessons and practice problems.