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-  Graph is from https://www.desmos.com/calculator
- Video from https://www.youtube.com/watch?v=Gfe2pnwivk4

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Featuring El Señor Olivares

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How to find real/imaginary solutions to polynomial functions:

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How to find real/imaginary solutions to polynomial functions:

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How To Find All Real and Imaginary Solutions or Zeros of Polynomial Functions

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Then after you get all the rational roots they need to plug in every positive and negative value of each root to see if the function equals 0. If it does equal zero, then that number is a factor of the function. So for the cubic function the factors would be: 3, 6 and 9. After you find a factor that equals zero, plug it into synthetic division to further break down the function into its factors. Once its into a quadratic, you can factor by unfoiling or simply by using the quadratic equation to get the rest of the factors.

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h(x)= x^4-4x^3+x^2+16x-20
h(x)=
(x-2+i)(x-2-i)(x-2)(x+2)
The Leading Term Test works for any function. The first term of a function determines its end behavior, and the terms following, determine its shape (i.e. hills and valleys). For this particular function, x^4 will have a specific end behavior (arrow up, arrow up). Thus since the function is an x^4 the end behavior will be the same, just with a different overall shape than x^4.
End Behavior:
Lim f(x)= ∞
x → ∞
Lim f(x)=∞
x → - ∞

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With the Rational Roots Theorem, mathematicians will be able to solve the expanded function into its factors. Using the P/Q (Factors of last term divided by factors of the first term) with plus and minuses the possible rational roots would be:
±(1,2,4,5,10,20) All factors of 20 (P, last term)
➗
± 1
All factors of 1 (Q, first term
After you listed all the possible factors, you need to separate them into the possible rational roots, by putting every number on top on every number on the bottom, for example: 1➗1= 1;  2➗1= 2;  4➗1= 4 (etc etc.. going down the list).  Now all your possible rational roots are:
±(1,2,4,5,10,20)

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Then after you get all the rational roots they need to plug in every positive and negative value of each root to see if the function equals 0. If it does equal zero, then that number is a factor of the function. So for the quartic function the factors would be: 2,-2, 2-i, and 2+i. After you find a factor that equals zero, plug it into synthetic division to further break down the function into its factors.  It's a lot easier to use rational numbers to find the first factors. Once its into a quadratic, you can factor by unfoiling or simply by using the quadratic equation to get the rest of the irrational factors. Or you can ever do synthetic division with one of the irrational roots if it's given to you to solve (see below).

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f(x)= x^3-18x^2+99x-162
f(x) = (x-3)(x-6)(x-9)
The Leading Term Test works for any function. The first term of a function determines its end behavior, and the terms following, determine its shape (i.e. hills and valleys). For this particular function, x^3 will have a specific end behavior (arrow down, arrow up). Thus since the function is an x^3 the end behavior will be the same, just with a different overall shape than x^3.
End Behavior:
Lim f(x)= ∞
x → ∞
Lim f(x) = - ∞
x → - ∞

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With the Rational Roots Theorem, mathematicians will be able to solve the expanded function into its factors. Using the P/Q (Factors of last term divided by factors of the first term) with plus and minuses the possible rational roots would be:
±(1,2,3,6,9,18,27,54,81,162) All factors of 162 (P, last term)
➗
± 1
All factors of 1 (Q, first term
After you listed all the possible factors, you need to separate them into the possible rational roots, by putting every number on top on every number on the bottom, for example: 1➗1= 1;  2➗1= 2;  3➗1= 3 (etc etc.. going down the list).  Now all your possible rational roots are:
±(1,2,3,6,9,18,27,54,81,162)