What is the number of turning points in the graph of the function of x defined below?

y = 2x² + 5x - 7

Solution:

The polynomial y has degree n=2 so the maximum number of turns it can have is n-1 = 2-1 = 1. (In fact we know that our polynomial is a quadratic and it opens upwards so this is another way to know it only has one turn).

The polynomial equation x(x2+4)(x2-x-6)=0 has how many real roots?

Solution:

x2+4 contributes complex zeros. This means that the factors x and x2-x-6 contribute real zeros. Therefore the polynomial has 3 real zeros

Find the inverse

Find the inverse of the 2x2 matrix below

[-3 1]
[-6 2]


Solution:

so our matrix is not invertible

Find exact value cos(2x) = 0 on the interval [0, 2pi]

how can i decompose this partial fraction(4x^3)/(x^3-x^2)

This is not a "proper" decomposition problem. Typically the degree of the denominator is greater than the degree of the numerator...

but...we can still simplify...
notice that this expression can be factored:

from here we notice that by polynomial long division (don't know how to show using this editor) that:

so our expression simplifies to...



:)

Write (2-2i)^8 in the standard form a+bi

Find all real numbers that satisfy the equation. cos 2x = sqrt(3)/2

or


where n is an integer, then...

let cos x = 1/sqrt(10) with x in quadrant IV. find cos 2x.

Using a double angle identity:

so we can write:

but cosine is positive in Quad IV, so...

Solve for all solution: sin(3x)cos(6x)-cos(3x)sin(6x)=-0.9

Use angle subtraction identity to get:
 

then since sine is an odd function



which leads to

z= ((3+4i)(12-5i))/(2+i) a. |z| ? b. Re (z) and Im (z)?

Expanding the numerator







 

        

In OPQ, o = 870 inches, P = 80 degrees, and Q = 9 degrees. Find the length of p to the nearest 10th of an inch.

We'll denote the measure of angle X as

In any triangle the sum of the angles is 180 degrees, so

so we know

Now using the Law of Sines we write:



which leads to

The function f(x)=2x/3x-5 is one-to-one. find its inverse and check your answer

To find the inverse we can solve for x to convert it to the dependent variable:










Once their roles have reversed, the resulting expression is the inverse:

To check we take the two appropriate inverses to see if they each recover the input (i.e. x).

Checking...

solve the equation ln(x)- ln (x-3) = ln(5)

combining logs...



raising e to both sides of the equation...

which leads to...




and so...