Polynomial Functions

A polynomial in one variable is a sum of terms of the form $a_n x^n$, where each exponent is a whole number.

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• Degree: the greatest exponent on the variable.
• Leading coefficient: the coefficient $a_n$ of the highest-degree term.
• Standard form: terms written from highest degree to lowest.
• Constant term: $a_0$, the term with no variable.

Classify by degree: $0$ constant, $1$ linear, $2$ quadratic, $3$ cubic, $4$ quartic, $5$ quintic.

The end behavior of a graph describes what happens to $f(x)$ as $x\to+\infty$ and $x\to-\infty$. It depends only on the degree parity and the sign of the leading coefficient.

Add / subtract: combine like terms. When subtracting, distribute the $-$ sign to every term first. Multiply: distribute each term of one factor across the other, then combine like terms. The degrees add: $(\deg m)\cdot(\deg n)$ product has degree $m+n$.

Long division works for any divisor. Synthetic division is a fast shortcut that works only when dividing by a linear factor $x-c$. The Division Algorithm says

Remainder Theorem: When $f(x)$ is divided by $x-c$, the remainder equals $f(c)$. Factor Theorem: $(x-c)$ is a factor of $f(x)$ if and only if $f(c)=0$ (that is, $c$ is a zero).

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• GCF first: always pull out the greatest common factor.
• Grouping: for four terms, group in pairs and factor each.
• Sum of cubes: $a^3+b^3=(a+b)(a^2-ab+b^2)$.
• Difference of cubes: $a^3-b^3=(a-b)(a^2+ab+b^2)$.
• Quadratic form: treat $x^4-5x^2+4$ like $u^2-5u+4$ with $u=x^2$.

Rational Root Theorem: any rational zero of a polynomial with integer coefficients has the form $\dfrac{p}{q}$, where $p$ divides the constant term and $q$ divides the leading coefficient. Strategy: test candidates until one works, use synthetic division to depress the polynomial to a lower degree, then repeat or factor the result. Multiplicity: the number of times a factor repeats. A zero of multiplicity $2$ makes the graph touch (bounce off) the $x$-axis; odd multiplicity makes it cross.

Fundamental Theorem of Algebra: every polynomial of degree $n\ge1$ has exactly $n$ zeros in the complex numbers, counted with multiplicity. Complex Conjugate Theorem: if the coefficients are real and $a+bi$ is a zero, then its conjugate $a-bi$ is also a zero. Complex zeros always come in pairs, so a polynomial of odd degree with real coefficients has at least one real zero.

To sketch $y=f(x)$:

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• Find the end behavior from degree and leading coefficient.
• Find the zeros (x-intercepts) and their multiplicities: cross at odd, touch at even.
• Plot the y-intercept $f(0)$.
• A degree-$n$ polynomial has at most $n-1$ turning points.
• Connect smoothly---polynomial graphs have no breaks or sharp corners.

For a monic polynomial with roots $r_1,\dots,r_n$ (repeated by multiplicity), expanding $\prod (x-r_i)$ shows that the elementary symmetric functions of the roots are the coefficients, with alternating signs. For a cubic $x^3+bx^2+cx+d$ with roots $r_1,r_2,r_3$:

For a quartic $x^4+bx^3+cx^2+dx+e$ with roots $r_1,r_2,r_3,r_4$:

Non-monic: for leading coefficient $a_n$, divide through first---each symmetric function is $(-1)^k a_{n-k}/a_n$.

Let $p_k=r_1^k+\cdots+r_n^k$ be the $k$th power sum of the roots and let $e_k$ be the $k$th elementary symmetric function (so $e_1=\sum r_i$, $e_2=\sum_{i<j} r_ir_j$, …). Newton's identities convert between them:

These let you compute $\sum r_i^k$ from the coefficients alone---no roots needed.

Dividing $P(x)$ by a quadratic leaves a remainder of degree at most $1$, say $R(x)=mx+k$. Since $P(x)=(x-a)(x-b)Q(x)+mx+k$, substituting the two roots kills the quotient term:

Solve this $2\times2$ linear system for $m$ and $k$. The remainder is then

a form of linear interpolation through the points $(a,P(a))$ and $(b,P(b))$.

Tabulate a degree-$n$ polynomial at equally spaced inputs and take successive differences of the outputs. The $n$th differences are constant and equal $a_n\cdot n!$ (for step size $1$), while the $(n+1)$th differences are $0$. This both detects the degree and reconstructs the leading coefficient.

More generally, $n+1$ data points determine a unique polynomial of degree $\le n$ (interpolation).

When a problem promises the roots follow a pattern, choose symmetric variables so Vieta collapses nicely.

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• Arithmetic (three roots): write them $r-d,\ r,\ r+d$. Their sum is $3r$, so $r$ equals $-\tfrac13(\text{coeff. of }x^2)$ immediately---the middle root is forced by the sum.
• Geometric (three roots): write them $\dfrac{r}{s},\ r,\ rs$. Their product is $r^3$, so $r$ is the cube root of the product of the roots.

Three tools sharply narrow the candidate list before you test anything.

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• Rational Root Theorem: every rational root is $\pm\dfrac{p}{q}$ with $p\mid a_0$, $q\mid a_n$.
• Upper/lower bounds (synthetic division): if you divide by $x-c$ with $c>0$ and the bottom row is all nonnegative, then $c$ is an upper bound---no root exceeds it. If you divide by $x-c$ with $c<0$ and the bottom row alternates in sign, then $c$ is a lower bound.
• Descartes' Rule of Signs: the number of positive real roots equals the number of sign changes in $P(x)$, or less by an even number. Apply to $P(-x)$ for the count of negative real roots.