Use the Remainder Theorem to find the remainder of (x^3-5x^2+7x+3)/(x-2). State whether the binomial is a factor of the polynomial.
Answer: Remainder = 1, not a factor
2) Determine the slant asympotte for f(x)=(x^2-5x-3)/x
Answer: y=x+5
1) I got a remainder of 5, check your work
2) correct
which of the following results is the graph of f(x)=x^2 being expanded vertically by factor of 3 and reflected over the x-axis
To use the Remainder Theorem to find the remainder of a polynomial divided by a binomial, we can substitute the value of the binomial into the polynomial and evaluate the result. The remainder will be the constant term that remains.
In this case, we have (x^3 - 5x^2 + 7x + 3) divided by (x - 2). To find the remainder, substitute x = 2 into the polynomial:
(2^3 - 5 * 2^2 + 7 * 2 + 3) = (8 - 20 + 14 + 3) = 1
Therefore, the remainder is 1.
To determine whether the binomial (x - 2) is a factor of the polynomial, we can use the fact that if the remainder is 0, then the binomial is a factor. In this case, since the remainder is 1, the binomial (x - 2) is not a factor of the polynomial.
Moving on to the second question:
To determine the slant asymptote of a rational function, we need to look at the degree of the numerator and the degree of the denominator.
In the given function f(x) = (x^2 - 5x - 3) / x, the degree of the numerator is 2 and the degree of the denominator is 1.
When the degree of the numerator is one degree higher than the degree of the denominator, we have a slant asymptote. The equation of the slant asymptote can be found by performing polynomial long division.
Performing long division:
x - 5
-------------
x | x^2 - 5x - 3
- (x^2 - 0x)
----------------
-5x - 3
+ 5x
-----------------
-3
The quotient is x - 5 and the remainder is -3. Therefore, the equation of the slant asymptote is y = x - 5.