**The fundamental theorem of algebra** any polynomial of degree n has n roots.

^{3}+ 3x

^{2}+ 7x + 9 then it is a polynomial at degree 3, and according to the fundamental theorem of algebra, will have 3 roots.

_{1})(x − r

_{2})(x − r

_{3})···

_{1}, r

_{2}, r

_{3}, …, are all roots.

Note: some polynomials will have “double” or “triple” roots. Some may also have complex roots. Therefore a polynomial of degree 4 can have 4 real roots (of which 2, 3 or 4 could be the same) or 4 complex roots (of which 2, 3 or 4 could be the same) or 2 real and 2 complex roots.

^{2}− 3x + 4 has complex roots, i.e. (b

^{2}− 4ac) < 0. The roots are

^{2}is a polynomial of degree 2, so it has 2 roots. However, the only root is x = 0. It means that x = 0 is a double root, meaning that the graph has a local minimum (or maximum) at that point.

_{c})

^{d}where r

_{c}is a root and d is a number of occurrences of that root.

^{5}+ x

^{4}− 8x

^{3}+ ax

^{2}+ bx + 24 with real coefficients a and b. It is also known that f(x) has a root x = −1 + i and one local minimum at x > 0.

x + 1 = ±i

(x + 1)^{2} = −1

x^{2} + 2x + 2 = 0

1. Since we know one factor of f (x), we can perform polynomial division to find other roots.

−2a − 12 = 0

a = −6

b + 16 − 2a − 36 = 0

b + 16 + 12 − 36 = 0

b = 8

x^{3} − x^{2} − 8x + (a + 18) = x^{3} − x^{2} − 8x + 12

^{3}+ bx

^{2}+ cx + d , one of the roots is usually some factor of

^{d}/

_{a}, so that it would be a rational fraction. As an example, our possible roots here are: ±12, ±6, ±4, ±3, ±2, ±1. By trying different values, you can find that x = 2 is one of our roots. Then it is required to perform polynomial division once again to get:

^{3}− x

^{2}− 8x + 12)÷(x − 2) = x

^{2}+ x − 6 = (x + 3)(x − 2)

^{2}+ 2x + 2)(x − 2)

^{2}(x + 3)

−1 + i − 1 − i + β + α + α = −^{1}/_{1}

2α + β = 1

(−1 + i)(−1 − i) × β × α × α = (−1)^{5} × ^{24}/_{1}

2 × β × α × α = −24

β × α^{2} = −12

^{2}+ 2x + 2)(x − 2)

^{2}(x + 3)