Cubic factoring is very simple on the science bowl questions and just refers to factoring cubic functions. Here is an easy example:
$$ \text{Find the roots of} \space4x^3-8x^2+6x-12 $$
Usually, the cubic factoring questions are rare and usually just involve grouping. Here, grouping can be done by clumping the common terms of different pairs of terms:
$$ 4x^2(x-2)+6(x-2) = (4x^2+6)(x-2)=0 \rightarrow x=2 $$
As can be seen above, the pair 4x^3 and -8x^2, and the pair 6x and -12 were factored together. Then, since the common term x-2 was present in both, both the expressions could be added because the x-2 was factored out. Also, in the end, 4x^2+6 does not have a 0 because the roots are imaginary.
Vieta's theorem is a theorem used to find the product/sum of the roots of any polynomial given the polynomial itself.
The product of the roots for a quadratic, the most basic polynomial, is c/a and is -b/a for the sum. The main pattern to take away is that the product of the roots is always the cornermost constant value divided by a, the coefficient of the term with the degree. For example, c is the constant in thsi case and a is the coefficient of x^2. which is the degree.
Now, let's see how this would play out for a polynomial with four roots (quartic), though a cubic polynomial is a maximum that science bowl will give you guys.
$$ y=ax^4+bx^3+cx^2+dx+e $$
Assume the roots of the polynomial are w, x, y, and z. Then:
$$ w+x+y+z=-\frac{b}{a} $$
$$ wx+wy+wz+xy+xz+yz=\frac{c}{a} $$
$$ wxy+wxz+wyz+xyz=-\frac{d}{a} $$
$$ wxyz=\frac{e}{a} $$
<aside> 💡 Note that "a" is the coefficient of the degree, which is 4 in this case. Also, note that "e" is the constant term at the end of the function.
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<aside> 💡 Note that the sign always alternates. Remember that the sum of the roots for ANY polynomial is ALWAYS -b/a.
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