write the types of rational fractions
Answers
Step-by-step explanation:
Rational functions
A rational function is a function of the form
f(x) = p(x)
q(x)
where p(x) and q(x) are polynomials in x with q ≡ 0. For example
x + 3
x − 7
,
x − 2
2x
3 + x
2 − x
,
x
2 + 3x + 2
1
.
The last is the same as x
2 + 3x + 2 , so any polynomial is also a rational function.
If the numerator and denominator have a common factor, we can simplify the fraction by
dividing top and bottom by that factor. For example,
x
2 + 3x + 2
x
2 + 2x + 1
=
(x + 1)(x + 2)
(x + 1)2
=
x + 2
x + 1
.
To multiply two rational functions, their numerators are multiplied together and their
denominators are multiplied together. To divide two rational functions, turn the second
one upside-down and multiply. For example,
4(x + 7)
x + 1
÷
x
2 + 5
2x + 2
=
4(x + 7)
x + 1
×
2x + 2
x
2 + 5
=
4(x + 7)(2x + 2)
(x + 1)(x
2 + 5) =
8(x + 7)
x
2 + 5
.
To add or subtract two rational functions, you must write them using a common denom-
inator. For example,
1
x + 1
+
2
x + 2
=
x + 2
(x + 1)(x + 2) +
2(x + 1)
(x + 1)(x + 2)
=
x + 2 + 2(x + 1)
(x + 1)(x + 2) =
3x + 4
(x + 1)(x + 2) .
Note. Often the common denominator is the product of the denominators, but sometimes
you can take something smaller. For example,
2
x + 1
−
x
(x + 1)(x + 2) =
2(x + 2)
(x + 1)(x + 2) −
x
(x + 1)(x + 2)
=
2(x + 2) − x
(x + 1)(x + 2) =
x + 4
(x + 1)(x + 2)
Step-by-step explanation:
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