Video: Properties of Rational Functions

Answer the following questions for the rational expressions (π‘₯ + 3)/3 and (π‘₯ βˆ’ 8)/2π‘₯. (i) Find the sum of (π‘₯ + 3)/3 and (π‘₯ βˆ’ 8)/2π‘₯. (ii) Is the sum of (π‘₯ + 3)/3 and (π‘₯ βˆ’ 8)/2π‘₯ a rational expression? (iii) Would this be true for any two rational expressions summed together?

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Video Transcript

Answer the following questions for the rational expressions π‘₯ plus three divided by three and π‘₯ minus eight divided by two π‘₯. One: find the sum of π‘₯ plus three divided by three and π‘₯ minus eight divided by two π‘₯. Two: is the sum of π‘₯ plus three divided by three and π‘₯ minus eight divided by two π‘₯ a rational expression? Three: would this be true for any two rational expressions summed together?

The word sum means add. Therefore, we need to add the two rational expressions or fractions π‘₯ plus three divided by three plus π‘₯ minus eight divided by two π‘₯. In order to add these two rational expressions, we’re going to follow four steps. Firstly, we will find the lowest common denominator. Secondly, we’ll rewrite each expression using the lowest common denominator. Thirdly, we will add the numerators. And lastly, we will simplify the expression as needed.

The lowest common denominator in this case is six π‘₯ as three multiplied by two π‘₯ is equal to six π‘₯. This means that we need to multiply the first expression by two π‘₯ and the second expression by three. Multiplying the first expression by two π‘₯ gives us two π‘₯ multiplied by π‘₯ plus three divided by six π‘₯. Multiplying the second expression by three gives us three multiplied by π‘₯ minus eight divided by six π‘₯. As the denominator is now the same, we can add the numerators. This leaves us two π‘₯ multiplied by π‘₯ plus three plus three multiplied by π‘₯ minus eight divided by six π‘₯.

Our next step is to multiply out the parentheses or brackets to simplify the expression. Two π‘₯ multiplied by π‘₯ is two π‘₯ squared and two π‘₯ multiplied by three is six π‘₯. Expanding the second bracket three multiplied by π‘₯ gives us three π‘₯ and three multiplied by negative eight gives us negative 24.

Finally, we can group the six π‘₯ and three π‘₯ on the numerator. This gives us a final answer of two π‘₯ squared plus nine π‘₯ minus 24 divided by six π‘₯. Therefore, the sum of π‘₯ plus three divided by three and π‘₯ minus eight divided by two π‘₯ is two π‘₯ squared plus nine π‘₯ minus 24 divided by six π‘₯.

Part two of the question asked, is the sum of π‘₯ plus three divided by three and π‘₯ minus eight divided by two π‘₯ a rational expression? Well, this is in effect asking, is our answer to part one two π‘₯ squared plus nine π‘₯ minus 24 divided by six π‘₯ a rational expression? Well, a rational expression is a fraction in which the numerator and denominator are polynomials.

In this case, the numerator two π‘₯ squared plus nine π‘₯ minus 24 is a quadratic. Therefore, it is a polynomial. The denominator six π‘₯ is a linear expression. It too is a polynomial this time with degree one. Since two π‘₯ squared plus nine π‘₯ minus 24 divided by six π‘₯ is a rational expression, then the sum of π‘₯ plus three divided by three and π‘₯ minus eight divided by two π‘₯ must also be a rational expression.

The final part of the question asked, would this be true for any two rational expressions summed together? Well, the short answer is yes. When we add or sum any two rational expressions, the answer will also be a rational expression.

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