Explain why this statement is true: when simplifying rational expressions, we must cancel factors only, and not terms. Give an example using just numbers.
e.g.
(12 + 6)/3
= 6(2+1)/3
= 2(3)
= 6
common error is to divide the 3 into the 12 to get
4 + 6 , that would be an example of canceling terms
( 12 * 48 ) /( 6 * 24 ) = ( 2 * 2 ) /( 1 * 1 ) = 4 / 1 = 4.
1. I divided 12 by 6 and 6 by 6.
2. I divided 48 by 24 and 24 by 24.
3. I multiplied 2 by 2 and 1 by 1.
4. Results = 4.
Now I'm going to change the multipli-
cation signs to plus signs:
( 6 + 24 ) / ( 12 + 48 ) =
30 / 60 = 2 / 4 = 1 / 2.
I had to get rid of the plus signs
by adding or combining the terms.
I can't legally divide 12 by 6 or 48 by 24.
10 divied by 16 =
In order to understand why we can only cancel factors and not terms when simplifying rational expressions, let's start by defining what factors and terms are.
In an algebraic expression or equation, terms are the individual parts that are separated by addition or subtraction. They can be constants (numbers), variables, or a product of both. For example, in the expression 3x - 2y + 5, the terms are 3x, -2y, and 5.
On the other hand, factors are the quantities that are multiplied together to form a product. They can also be constants or variables, or a combination of both. For example, in the expression (x + 1)(x - 2), the factors are (x + 1) and (x - 2).
When simplifying rational expressions, we have to remember that they are essentially fractions in the form of one algebraic expression divided by another. For example, (x^2 + 3x + 2) / (x + 1) is a rational expression.
Now, let's consider why we can only cancel factors and not terms when simplifying rational expressions. The key point is that when we cancel factors, we are essentially dividing both the numerator and the denominator of the fraction by the same quantity. This is a valid operation in mathematics and does not change the value of the fraction.
However, if we were to cancel terms, we would be subtracting or adding quantities, which would alter the overall expression or equation. This would result in an incorrect simplification.
Here's an example using just numbers to illustrate this concept:
Consider the expression (3x + 6) / 3. If we want to simplify this expression, we can notice that both 3x and 6 have a common factor of 3. Thus, we can cancel out the common factor:
(3x + 6) / 3 = (3 * x + 3 * 2) / 3 = (3 * (x + 2)) / 3
Now, we can cancel the common factor of 3 in numerator and denominator:
(3 * (x + 2)) / 3 = (cancel out 3) = x + 2
In this example, we only canceled the factor of 3 and not the term "x" in the numerator because canceling the term "x" would change the entire meaning of the expression.
Therefore, when simplifying rational expressions, we must always remember to cancel only common factors and not terms to ensure an accurate simplification.