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Question Video: Subtraction of Rational Expressions Where the Denominators Can Be Factored Mathematics

Simplify ((5π‘₯ βˆ’ 2)/3π‘₯Β²) βˆ’ ((7π‘₯ βˆ’ 2)/9π‘₯⁴).

02:53

Video Transcript

Simplify five π‘₯ minus two over three π‘₯ squared minus seven π‘₯ minus two over nine π‘₯ to the power of four.

For this question, we have been asked to simplify the difference of two rational expressions. Remember that a rational expression is the quotient of two polynomials. We can think of these expressions as algebraic fractions. And hence when trying to simplify sums and differences of rational expressions, there are many similarities to the methods used for numerical fractions.

For instance, the first step that we’ll need to perform is expressing both of our terms with the same denominator. One way to do this is to multiply the top and bottom of our first term by the denominator of our second term, which is nine π‘₯ to the power of four. We then multiply the top and bottom of our second term by the denominator of our first term, which is three π‘₯ squared. Doing so would give us a common denominator of 27π‘₯ to the power of six for both of our terms. Although this would be a perfectly reasonable method, we can actually cut down on our calculations by first considering whether our denominators have a common factor. Let’s clear some room and take a step back.

Looking at our two denominators, we can see that nine π‘₯ to the power of four is equal to three π‘₯ squared times three π‘₯ squared. This means that yes, our two denominators do share a common factor of three π‘₯ squared. If we multiply the top and bottom of our first term by three π‘₯ squared, it will match the denominator of our second term. After doing so, both terms will share the same denominator of nine π‘₯ to the power of four. And this is exactly the condition that we need to continue with our simplification. Five π‘₯ minus two multiplied by three π‘₯ squared is 15π‘₯ to the power of three minus six π‘₯ squared. This is the numerator of our first term. The denominator is, of course, nine π‘₯ to the power of four.

Now that both of our terms have the same denominator, we can express the given difference as a single algebraic fraction. Note that the final term of our numerator has a positive sign. Since we are subtracting a negative two, we are left with a positive two. In the resulting algebraic fraction, we cannot combine any of the terms in our numerator, nor can we cancel any common factors. Since there are no more useful simplifications, this is our answer. The simplified version of the given expression is 15π‘₯ cubed minus six π‘₯ squared minus seven π‘₯ add two over nine π‘₯ to the power of four.

As a final note, our question did not ask us to consider this rational expression as a function. If it had, we might need to include extra steps early on in our method to consider the domain of the function. Considering the domain is not relevant to this question, but it is definitely worth looking out for in the future.

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