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''This subsection is a review only and we take the main results as known. For proofs, see {{harv|Birkhoff|MacLane|1965}} or {{harv|Ebbinghaus|1990}}.''

Just as integers have a division operation&mdash; e.g.,
"<math> 4 </math> goes <math> 5 </math> times into <math> 21 </math> with remainder <math> 1 </math>"&mdash; so
do polynomials.

{{TextBox|1=
;Theorem 1.1 (Division Theorem for Polynomials){{anchor|th:EuclidForPolys}}:<!--\label{th:EuclidForPolys}-->
Let <math> c(x) </math> be a polynomial.
If <math> m(x) </math> is a non-zero polynomial then there are '''quotient''' and
'''remainder''' polynomials <math> q(x) </math> and <math> r(x) </math> such that

:<math>
c(x)=m(x)\cdot q(x)+r(x)
</math>

where the degree of <math> r(x) </math> is strictly less than the degree of <math> m(x) </math>.
}}

In this book constant polynomials,
including the zero polynomial, are said to have degree <math> 0 </math>.
(This is not the standard definition, but it is convienent here.)

The point of the integer division statement
"<math> 4 </math> goes <math> 5 </math> times into <math> 21 </math> with remainder <math> 1 </math>"
is that the remainder is less than <math> 4 </math>&mdash; while <math> 4 </math> goes <math> 5 </math>
times, it does not go <math> 6 </math> times.
In the same way, the point of the polynomial division statement
is its final clause.

{{TextBox|1=
;Example 1.2:
If <math> c(x)=2x^3-3x^2+4x </math> and <math> m(x)=x^2+1 </math> then <math> q(x)=2x-3 </math> and <math> r(x)=2x+3 </math>. Note that <math> r(x) </math> has a lower degree than <math> m(x) </math>.
}}

{{TextBox|1=
;Corollary 1.3:
The remainder when <math> c(x) </math> is divided by <math> x-\lambda </math> is the constant polynomial <math> r(x)=c(\lambda) </math>.
}}

{{TextBox|1=
;Proof:
The remainder must be a constant polynomial because it is of degree less than the divisor <math> x-\lambda </math>, To determine the constant, take <math>m(x)</math> from the theorem to be <math>x-\lambda</math> and substitute <math> \lambda </math> for <math>x</math> to get <math> c(\lambda)=(\lambda-\lambda)\cdot q(\lambda)+r(x) </math>.
}}

If a divisor <math> m(x) </math> goes into a dividend <math> c(x) </math> evenly,
meaning that <math> r(x) </math> is the zero polynomial,
then <math> m(x) </math> is a '''factor''' of <math> c(x) </math>.
Any '''root''' of the factor (any <math> \lambda\in\mathbb{R} </math> such that
<math> m(\lambda)=0 </math>) is a root of <math> c(x) </math> since
<math> c(\lambda)=m(\lambda)\cdot q(\lambda)=0 </math>.
The prior corollary immediately yields the following converse.

{{TextBox|1=
;Corollary 1.4:
If <math> \lambda </math> is a root of the polynomial <math> c(x) </math> then <math> x-\lambda </math> divides <math> c(x) </math> evenly, that is, <math>x-\lambda</math> is a factor of <math>c(x)</math>.
}}

Finding the roots and factors of a high-degree polynomial can be hard.
But for second-degree polynomials we have the quadratic formula: the
roots of <math> ax^2+bx+c </math> are

:<math>
\lambda_1=\frac{-b+\sqrt{b^2-4ac}}{2a}
\qquad
\lambda_2=\frac{-b-\sqrt{b^2-4ac}}{2a}
</math>

(if the discriminant <math> b^2-4ac </math> is negative then the polynomial
has no real number roots).
A polynomial that cannot be factored into two lower-degree polynomials
with real number coefficients is '''irreducible over the reals'''.

{{TextBox|1=
;Theorem 1.5:
Any constant or linear polynomial is irreducible over the reals. A quadratic polynomial is irreducible over the reals if and only if its discriminant is negative. No cubic or higher-degree polynomial is irreducible over the reals.
}}

{{TextBox|1=
;Corollary 1.6:
Any polynomial with real coefficients can be factored into linear and irreducible quadratic polynomials. That factorization is unique; any two factorizations have the same powers of the same factors.
}}

Note the analogy with the prime factorization of integers.
In both cases, the uniqueness clause is very useful.

{{TextBox|1=
;Example 1.7:
Because of uniqueness we know, without multiplying them out, that <math> (x+3)^2(x^2+1)^3 </math> does not equal <math> (x+3)^4(x^2+x+1)^2 </math>.
}}

{{TextBox|1=
;Example 1.8:
By uniqueness, if <math> c(x)=m(x)\cdot q(x) </math> then where <math> c(x)=(x-3)^2(x+2)^3 </math> and <math> m(x)=(x-3)(x+2)^2 </math>, we know that <math> q(x)=(x-3)(x+2) </math>.
}}

While <math> x^2+1 </math> has no real roots and so doesn't factor over the real
numbers, if we imagine a root&mdash; traditionally denoted <math> i </math>
so that <math> i^2+1=0 </math>&mdash; then <math> x^2+1 </math> factors into a product of linears
<math> (x-i)(x+i) </math>.

So we adjoin this root <math> i </math> to the reals and close the new system with
respect to addition, multiplication, etc. (i.e., we also add
<math> 3+i </math>, and <math> 2i </math>, and <math> 3+2i </math>, etc., putting in all linear
combinations of <math>1</math> and <math>i</math>).
We then get a new structure, the '''complex numbers''', denoted
<math> \mathbb{C} </math>.

In <math>\mathbb{C}</math> we can factor (obviously, at least some) quadratics that would be
irreducible if we were to stick to the real numbers.
Surprisingly, in <math> \mathbb{C} </math> we can not only factor <math> x^2+1 </math>
and its close relatives, we can factor any quadratic.

:<math>
ax^2+bx+c=
a\cdot \big(x-\frac{-b+\sqrt{b^2-4ac}}{2a}\big)
\cdot \big(x-\frac{-b-\sqrt{b^2-4ac}}{2a}\big)
</math>

{{TextBox|1=
;Example 1.9:
The second degree polynomial <math> x^2+x+1 </math> factors over the complex numbers
into the product of two first degree polynomials.

:<math>
\big(x-\frac{-1+\sqrt{-3}}{2}\big)
\big(x-\frac{-1-\sqrt{-3}}{2}\big)
=
\big(x-(-\frac{1}{2}+\frac{\sqrt{3}}{2}i)\big)
\big(x-(-\frac{1}{2}-\frac{\sqrt{3}}{2}i)\big)
</math>
}}

{{TextBox|1=
;Corollary 1.10 (Fundamental Theorem of Algebra):
Polynomials with complex coefficients factor into linear polynomials with complex coefficients. The factorization is unique.
}}
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== References ==
* {{citation|first1=H. D.|last1=Ebbinghaus|title=Numbers|publisher=Springer-Verlag|year=1990}}.
* {{citation|first1=Garrett|last1=Birkhoff|first2=Saunders|last2=MacLane|title=Survey of Modern Algebra|edition=Third|publisher=Macmillian|year=1965}}.

{{navigation|Book=Linear Algebra
|current=Factoring and Complex Numbers: A Review
|previous=Complex Vector Spaces
|next=Complex Representations}}

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