Question

From where I left off this is what is left... 3*(x+1)*x=90

Simplify the equation and write it in standard form.
Find the solutions to this equation algebraically using the Fundamental Theorem of Algebra, the Rational Root Theorem, Descartes' Rule of Signs, and the Factor Theorem.

(Hint: If the numbers are large, graph the function first using GeoGebra to help you find one of the zeros. Use that zero to find the depressed equation which can be solved by factoring or the quadratic formula.)

Substitute 0 for the function notation and, using graphing technology, graph the function.

Answer 1

@experimentX

Answer 2

@hartnn

Answer 3

\[3(x+1)x = 90\]\[3(x²+x) = 90\]\[x² + x = 30\]\[x² + x - 30 = 0\]
The last line is the standard form. I don't know the Descartes Rules of Signs and I'm not sure what the Factor Theorem is, but I think I can do the other stuff.

The Fundamental Theorem of Algebra tells you that there are 2 different roots to this polynomial as the highest occurring potency is 2.

The Rational Root Theorem tells you that for any rational root the root must be a factor of 30 (it's always the number that is not a coefficient). This means we can just try out some factors of 30 and see if they're roots.

In this particular example, we see that 5 is a factor of 30 which satisfies the equation as 5² + 5 = 30.

Now, executing a polynomial divison we get x²+x-30 : (x-5) = (x+6).
This is equivalent to (x-5) * (x+6) = x² + x - 30.
(I think i used the Factor Theorem there)
Now we can see at one glance that the zeroes of this polynomial are equal to the zeroes of (x-5) * (x+6), which obviously are x=5 and x= -6. Because of the Fundamental Theorem of Algebra we know that we've found all the zeroes because there are exactly 2 of them.

Answer 4

Wow honestly I really appreciate the help guys x.x

Answer 5

Factor theorem says that
\[ f(x) = x²+x−30\]
Keep putting the values of 'x', for what value of 'x' (say 'a') you get zero, (x-a) will be factor of f(x).

\[ f(x) = x²+x−30\]
+ + -
one sign change here, so it says that one root will be positive
\[ f(-x) = x²-x−30\]
+ - -
one sign change here, so it says that one root will be negative.
From fundamental theorem of algebra, you have 2 roots, .. no all real roots ,,, no complex root.

Answer 6

Thank you experiment you guys are the best!