Y-Intercept: The constant term in the polynomial equation is the Y-intercept. The y-intercept for this equation is (0,30). This is one of the points that is easily spotted on the graph.
Descarte's Rule of signs: This rule allows you to how many positive, negative, or imaginary solutions there are.
First, count the number of sign changes (+/-) are in the original equation. This will tell you how many positive solutions there are. To find out how many negative solutions there are, multiply the equation by -1 and count the sign changes. An odd degree will change the sign to opposite, an even degree will leave it the same.
f(x)= 7x^3 + 12x^2 -109x + 30
Positive: + , + , - , + 2 sign changes
Negative: -7x^3 - 12x^2 + 109x + 30 1 sign change
Imaginary : There are three roots, and 2 are positive and one is negative, so there are no imaginary roots.
Rational Root Theorem: Find the factors of the constant term and the leading coefficient.
30: +/- for each... 1, 2, 3, 5, 6, 10, 15, 30
7: +/- for each... 1, 7
Then put the factors of the top number over the bottom.
Other possible factor could include: +/- 2/7. 3/7, 5/7, 6/7, 10/7, 15/7, 30/7
Synthetic Division: - 5 | 7 12 -109 30
|___ -35__ 115_-30_____
3 | 7 -23 6 0
|____21___-6_____________
7 -2 0
Place the coefficients of your equation inside of this figure, and leave space below. Place one of the possible solutions outside of the figure. Bring down the leading coefficient to start.
Step 1: Multiply the solution by the coefficient. Place your answer under the next coefficient, then add. Repeat for all coefficients. If your last part adds to zero, the solution is correct, and the equation is now narrowed down a degree, repeat until all solutions are shown.
The solutions for this polynomial are -5, 3, 2/7
You can also plug your equation into a calculator to see the graph of it and know your solutions are correct.
On Descartes, does it HAVE to be two positive roots? Can you have 0?
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