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Polynomials and Roots

A polynomial is a linear expression of terms consisting of coefficients and variables whose powers are nonnegative integers. Polynomials may be single variate such as \(2x^3+3x+6\), or they may be multivariate such as \(x^2y^2+2xy^2-3x+5\).

The degree of a polynomial is the maximum sum of the exponents for each term in the polynomial. For instance, the degree of \(f(x,y)=x^2y^2+3x^2y+x^2\) is \(deg(fx,y)=4\) since \(2+2=4\) is greater than \(2+1=3\) and \(2\). The degree of a single variate polynomial corresponds to the highest power within the given polynomial.

The vertex form for the quadratic \(f(x)=ax^2+bx+c\) is of the form \(f(x)=a(x-h)^2+k\).

The vertex is the point \((h,k)\) on the quadratic.

The value \(y=k\) is maximum on the quadratic provided that \(a<0\), and \(y=k\) is a minimum if \(a>0\). The process of finding the x-coordinate of the vertex can be found completing the square. This value can also be found by noting that \(\displaystyle x=\frac{-b}{2a}\).

Classify Polynomials

Polynomials can be classified based upon the number of terms present within the polynomial.
Number of Terms Classification
One Term Monomial
Two Terms Binomial
Three Terms Trinomial
Four Terms Quadnomial
Polynomials can be classified based upon the highest degree present within the polynomial.
Highest Degree Classification
Degree One Linear
Degree Two Quadratic
Degree Three Cubic
Degree Four Quartic
Degree Five Quintic
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Polynomial

Binomial Theorem

\(\displaystyle (a+b)^n = \sum_{k=0}^{n} \left(\begin{array}{c}n\\k\end{array}\right) a^nb^{n-k} \)

where \(\displaystyle \left(\begin{array}{c}n\\k\end{array}\right) = \frac{n!}{(n-k)!k!} \)
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Quadratic Formula

Roots

The roots of the quadratic

\(f(x)=ax^2+bx+c\)

can be found by using the quadratic formula.

\(\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)
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Polynomial

Definition Roots

If \(f(x)\) is a polynomial and \(f(c)=0\) then the value \(c\) is called a root, or a zero.

\(f(c)=0\) if and only if the polynomial \((x-c)\) is a factor for the polynomial \(f(x)\)