Question

1. Are x^2-1 and tan x=sin x/cos x rational expressions ?
2. The number of excluded values of x^3+x^2-10x+8 / x^4+8x^2-9 is______________​

pl help fast dears.....

Answer 1

Answer:

1. $x^{2} -1$ is a rational expression but $tanx = \frac{sinx}{cosx}$ is not a rational expression

2. Number of excluded values = 2

Step-by-step explanation:

1. Rational expressions are fractions where numerator and denominator are polynomials. $x^{2} -1$ can be written as $\frac{x^{2}-1}{1}$. Here $x^{2} -1$ and 1 both are polynomials. Therefore $x^{2} -1$ is a rational expression.

$tanx$, $sinx$ and $cosx$ are trigonometric functions and therefore they are not polynomials.

2. In order to find the number of excluded values of $\frac{x^{3}+x^{2}-10x+8}{x^{4}+8x^{2}-9}$ we have to see for the values where the denominator is 0

Thus,

$x^{4} +8x^{2} -9=0$

or, $x^{4} +9x^{2}- x^{2} -9=0$

or, $x^{2} (x^{2} +9)-1(x^{2} +9)=0$

or, $(x^{2} +9)(x^{2} -1)=0$

This gives, $x^{2} =-9$, which is not possible for real values of x

and $x^{2} -1=0$

or, $(x-1)(x+1)=0$

This gives, $x=1, -1$

Thus, the number of excluded values of the given expression are 2, which are 1 and -1

Answer 2

Answer:

Step-by-step explanation:

Our questions are: 1. Are x^2-1 and tan x=sin x/cos x rational expressions ?  

2. The number of excluded values of x^3+x^2-10x+8 / x^4+8x^2-9 is______________​

In order to answer this questions I will answer them separately.

Question 1:

    Rational expressions are fractions where numerator and denominator are polynomials.

x^2 - 1 can be written as (x^2 - 1)/1. In this example we can say that x^2 - 1 and 1 are both polynomials. Therefore it is a rational expression.

Also there are a lot of expressions that are not rational expressions, like tanx, sinx, cosx are trigonometric functions and therefore they are not polynomials.

Question 2:

    Since we want to check the number of excluded values of that expression, we need to check when is the denominator equal to 0.

Hence we have x^4 + 8 * x^2 - 9 = 0.

That is equivalent to (x^2 + 9)(x^2 - 1) = 0 as you can check as an exercise.

From where we get x^2 = -9, which is impossible since the square of a negative number is not real.

And also x^2 = 1, and we get x = 1 or x = -1.

Hence, the number of excluded values of the initial expression are 1 and -1.

I hope this helps your studies!!

Keep it up!!