y=x^3+5x^2-3x-6
Descartes:
To find positive solutions, take the “symbol”, positive or negative, ahead of the coefficient. To find negative solutions, insert (-x) into the equation and repeat what you did for negative.
Positive: + + - -, 1 change indicates 1 possible positive solutions,
Negative: - + - -, 2 changes indicates 2 negative solutions or 0 possible solutions because imaginary roots come in pairs
Y- intercept
To find the y intercept, plug 0 in for x, or simply look at the constant, or where the function crosses the y axis on the graph.
The constant of this equation is -6, so that is your y-intercept
Rational Root Test
To find all possible solutions, (where the function crosses the x axis) put your constant over the leading coefficient and all of their factors.
6/1 = +/- (1,2,3,6)/(1) = +/- 1, 2, 3, 6
Synthetic Division
List the leading coefficient for each term, if a term is missing write "0". Draw a line under them leaving space for another row of numbers. Also draw a line up the left side of the new line. To the left of this new line, write in a possible solution (found in rational root test). Bring the first number down in its column below the bottom line. Multiply it to the possible root and place this new number in the second column under the second coefficient. Add these two numbers together and place that new number under the line in the second column and repeat. If the final number you get in the last column is a 0, the root is a solution to the function. If the division ends with 3 numbers left and no more solutions, make those numbers into a quadratic equation and factor to find i.
-1 |1 5 -3 -6
|_-2_-3_6__
1 3 -6 0
This proves that -1 is a solution to the function, to complete all other solutions, plug in the possible solution, guess and checking until you find the rest of the roots.
Using a calculator
To check your solutions using a calculator, put your equation in the "y=" section. Press graph and trace your solutions or check them in the data table.
Triangles
45/45/90 Triangle
Each leg has the measure of the square root of two and the hypotenuse measures at 2
sin(45)= (sq root 2)/2
cos(45)= (sq root 2)/2
tan(45)=1
60/30/90 Triangle
The side corresponding to the 60 degree angle has a measure of the square root of 3, the side length corresponding to the 30 degree angle is 1 and the hypotenuse is 2
sin(60)= (sq root 3)/2
cos(60)= 1/2
tan(60)= (sq root 3)
30/60/90 Triangle
The same as the 60/30/90 triangle, but turned sideways. All side lengths correspond to the same angles.
sin(30)=1/2
cos(30)=(sq root 3)/2
tan(30)=(sq root 3)/3
