Zeros of Polynomials: Chapter Review

Vocabulary and notation

Multiplicity, simple, repeated

The exponent $m$ in $p(x) = (x - r)^m q(x)$, $q(r) \neq 0$: how many times $(x - r)$ divides $p$. $m = 1$ is simple, $m \ge 2$ repeated.

Counted with multiplicity, versus distinct

The root list repeats a root as often as its multiplicity; the distinct roots are the values on it. $(x - 2)^3(x + 1)$: four with multiplicity, two distinct.

Complex conjugate $\overline{z}$

$\overline{a + bi} = a - bi$. Respects $+$ and $\times$, and fixes exactly the reals: $\overline{z} = z$ if and only if $z$ is real.

Surd partner $s^{*}$

$(a + b\sqrt{d})^{*} = a - b\sqrt{d}$, from the swap $\sqrt{d} \mapsto -\sqrt{d}$. A DIFFERENT swap from the bar, fixing exactly the rationals: $2 + \sqrt{5}$ is real, so $\overline{2 + \sqrt{5}}$ is itself.

Elementary symmetric function $e_k$

The sum of every product of $k$ different ENTRIES of the root list: $e_1$ the roots, $e_2$ all pairwise products, $e_n$ the product of all. Swapping two roots leaves it unchanged.

Depressed polynomial

The quotient left after dividing out a root, one degree lower. The remaining roots are hunted there.

Sign chart, test point

The real zeros cut the line into intervals; the chart records the sign $P$ holds on each. A test point is one value strictly inside an interval, never a zero.

Turning point

Where the graph stops rising and starts falling, or the reverse. It need not lie on the axis, and a zero need not be one.

Formulas and theorems

• Fundamental Theorem of Algebra, and complete factorization

  $$p(x) = a(x - r_1)(x - r_2)\cdots(x - r_n)$$

  Use when Degree $n \ge 1$; coefficients may be complex, as the induction needs. $a \neq 0$ is the leading coefficient; the $r_i$ are listed with multiplicity and are the ONLY roots. Existence only: it names no root, and for $n \ge 5$ no general radical formula exists. A constant such as $p(x) = 7$ has none.

• Root Counting Theorem

  $$m_1 + m_2 + \cdots + m_k = n$$

  Use when $s_1, \ldots, s_k$ the DISTINCT roots with multiplicities $m_i$, over $\mathbb{C}$, $n \ge 1$. So exactly $n$ roots counted with multiplicity, at most $n$ distinct, and exactly $n$ distinct precisely when every root is simple. The REAL count is unconstrained, from $0$ to $n$.

• Conjugate root theorem, and the real quadratic it forces

  $$\begin{gathered} p(z) = 0 \ \Longrightarrow \ p(\overline{z}) = 0 \\ (x - z)(x - \overline{z}) \\ = x^2 - 2ax + (a^2 + b^2) \end{gathered}$$

  Use when EVERY coefficient real, and $z = a + bi$ with $b \neq 0$ for the factor. Strip that and it dies: $x - i$ has root $i$ and not $-i$. Not reversible: $i(x^2 + 1)$ pairs $i, -i$ with nonreal coefficients. Real roots are their own conjugates; nonreal ones pair with EQUAL multiplicity. The factor's discriminant $-4b^2 < 0$ keeps it unsplittable over $\mathbb{R}$.

• Surd conjugate theorem, and its factor

  $$\begin{gathered} p\big(a + b\sqrt{d}\,\big) = 0 \\ \Longrightarrow \ p\big(a - b\sqrt{d}\,\big) = 0 \end{gathered}$$

  Use when Coefficients RATIONAL, strictly more than real; $a$, $b$, $d$ all rational, $b \neq 0$, $\sqrt{d}$ irrational. The pair forces the factor $x^2 - 2ax + (a^2 - b^2d)$. That $d$ is rational is load-bearing: with $d = 3 + 2\sqrt{2}$, so $\sqrt{d} = 1 + \sqrt{2}$, the root $1 + \sqrt{2}$ of $x^2 - 2x - 1$ has no partner $-1 - \sqrt{2}$.

  e.g. A root $2 + \sqrt{5}$ forces the factor $x^2 - 4x - 1$.

• Factoring a real polynomial over $\mathbb{R}$, and the parity it forces

  $$\begin{gathered} p(x) = a(x - r_1)\cdots(x - r_k) \\ \cdot\ d_1(x)\cdots d_m(x) \\ n = k + 2m \end{gathered}$$

  Use when Real coefficients, $n \ge 1$, each $r_i$ real and each $d_j$ a real quadratic of negative discriminant. The nonreal roots number $2m$, always EVEN, so $k$ and $n$ share a parity. Odd $n$ forces $k \ge 1$: at least one real root, possibly exactly one. A real cubic has $1$ or $3$ real roots WITH MULTIPLICITY, never $0$.

• Vieta's formulas

  $$e_k = (-1)^k\,\frac{a_{n-k}}{a_n}, \qquad k = 1, 2, \ldots, n$$

  Use when $a_n \neq 0$, and $r_1, \ldots, r_n$ the COMPLETE list over $\mathbb{C}$, repeats repeated and nonreal roots kept. The sign counts how many roots the term collects, not sum versus product: the product of the roots is $-\frac{a_0}{a_n}$ for a cubic but $+\frac{a_0}{a_n}$ for a quartic. A missing term means $a_j = 0$. Monic template: $x^3 - e_1x^2 + e_2x - e_3$.

  e.g. $(x - 1)(x - 2)(x - 3) = x^3 - 6x^2 + 11x - 6$, and $e_1 = 6$, $e_2 = 11$, $e_3 = 6$.

• Symmetric quantities built from the $e_k$

  $$\begin{gathered} r_1^2 + \cdots + r_n^2 = e_1^2 - 2e_2 \\ \frac{1}{r_1} + \cdots + \frac{1}{r_n} = \frac{e_{n-1}}{e_n} \end{gathered}$$

  Use when Any degree $n \ge 2$; the first holds at every degree. The second needs every root nonzero, failing exactly when $a_0 = 0$.

• Multiplicity decides how the graph meets the axis

  $$\begin{gathered} P(x) = (x - r)^m G(x), \ G(r) \neq 0 \\ m \ \text{odd} \iff \text{crosses} \\ m \ \text{even} \iff \text{touches and turns back} \end{gathered}$$

  Text description

  A curve passing straight through the axis at a zero of odd multiplicity, and meeting the axis at a zero of even multiplicity without crossing, turning back the way it came.

  Use when $P$ real, $r$ a REAL zero. Both run both ways, and $m = 1$ is odd, so a simple zero crosses. Even $m$ makes $(r, 0)$ a turning point. A larger $m$ only flattens the curve; it never changes which happens. Nonreal roots give no $x$-intercept: $-(x - 2)^2(x^2 + 1)$ has degree $4$ and one intercept.

  e.g. $(x - 3)^2(x + 1)$: $P(2) = 3$, $P(4) = 5$, so it touches at $3$; $P(-2) = -25$, $P(0) = 9$, so it crosses at $-1$.

• End behavior, all four cases

  $$P(x) \ \text{follows} \ a_nx^n \ \text{for large} \ |x|$$

  Text description

  Four miniature graphs in a grid: even degree sends both arms the same way, up for a positive leading coefficient and down for a negative one, while odd degree sends them opposite ways.

  Use when Real $P$ of degree $n \ge 1$, leading coefficient $a_n \neq 0$. Even $n$: both arms up if $a_n > 0$, both down if $a_n < 0$. Odd $n$: left down and right up if $a_n > 0$, left up and right down if $a_n < 0$.

• Sign changes, and which turning points are reachable

  $$\begin{gathered} P(u)\,P(v) < 0 \\ \Longrightarrow \ P(c) = 0 \ \text{for some} \ c \\ \text{between} \ u \ \text{and} \ v \end{gathered}$$

  Use when Real $P$, unbroken graph. The converse fails, an even zero having no sign change: a change REQUIRES a real zero, a real zero does not force one. So $P$ holds one sign on any interval free of real zeros, and one test point settles it. At most $n - 1$ turning points, at least one strictly between consecutive real zeros; their coordinates are beyond algebra apart from a parabola's vertex and the even zeros themselves.

Problem types, step by step

Exam traps

• Trap "A degree-$n$ polynomial has $n$ roots", so $x^2 - 2x + 1$ is given $2$ distinct roots and $x^2 + 1$ is given $2$ real ones.

  Fix $(x - 1)^2$ has one distinct root, and $x^2 + 1$ has no real root. Exactly $n$ roots, counted with multiplicity, over $\mathbb{C}$.

• Trap Pairing a nonreal root without checking the coefficients: reading $2(x - 3)^2(x + i)$ as having roots $3, 3, -i, i$.

  Fix That is degree $3$, and $i$ is not a root. The pairing needs EVERY coefficient real; expanded, this one carries $2i - 12$ and $18i$.

• Trap Using the surd pairing when the coefficients are merely real, or putting a bar on a surd.

  Fix $x^2 - (2 + \sqrt{2})x + 2\sqrt{2} = (x - 2)(x - \sqrt{2})$ is real with $\sqrt{2}$ a root and $-\sqrt{2}$ not. Real buys the complex pairing; the surd pairing costs rational.

• Trap Vieta from memory: the product of the roots of $x^4 - 3x^3 + 7x - 2$ read as $2$, or the sum for $2x^3 - 6x^2 + x - 5$ read as $6$.

  Fix The product is $-2$: the sign is $(-1)^k$ and $k = 4$ is even. The sum is $-\frac{-6}{2} = 3$: every symmetric sum divides by $a_n$.

• Trap Feeding Vieta an incomplete list: $x^3 - 5x^2 + 8x - 4 = (x - 1)(x - 2)^2$ summed as $1 + 2 = 3$.

  Fix The list is $1, 2, 2$, summing to $5$. Nonreal roots count too: $x^3 - 5x^2 + 8x - 6$ has list $3, 1 + i, 1 - i$, again $5$.

• Trap Treating every real zero as a crossing, so $(x - 3)^2(x + 1)$ is drawn punching through the axis at $3$.

  Fix That multiplicity is even, so it touches and turns back, and $(3, 0)$ is a turning point. Test points on both sides give the same sign.

• Trap Labelling a hump never evaluated, or placing it midway between two zeros.

  Fix For $(x + 2)(x - 1)^2(x - 3)$ the midpoint $P(-0.5) = -11.8125$ sits above $P(-1) = -16$, so the low point is elsewhere. Claim only evaluated heights.