Theory of Equations — Study Notes

This chapter is about relationships: how the coefficients of a polynomial equation encode its roots, how known roots let you build or shrink an equation, and what general rules (Fundamental Theorem of Algebra, conjugate-root theorems, Rational Root Theorem, Descartes’ Rule) tell you before you ever solve. Master the relationships and most problems become bookkeeping.

1. Polynomials and polynomial equations

A polynomial of degree \(n\) in one variable is \(a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}\) with \(a_{n}\neq 0\). Key vocabulary:

\(a_{n}\) is the leading coefficient and \(a_{n}x^{n}\) the leading term.
If \(a_{n}=1\) the polynomial is monic.
A value \(c\) with \(P(c)=0\) is a zero of the polynomial and a root of the equation \(P(x)=0\).
Degrees 2, 3, 4 are called quadratic, cubic, quartic.
Exponents must be non-negative integers — \(3x^{-2}+1\) and \(5x^{1/2}+1\) are not polynomials.

2. Quadratic recap

For \(ax^{2}+bx+c=0\) the roots are \(\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). The discriminant \(\Delta=b^{2}-4ac\) decides their nature when \(a,b,c\) are real:

\(\Delta\gt 0\): two real and distinct roots.
\(\Delta=0\): real and equal roots.
\(\Delta\lt 0\): no real roots (a conjugate pair of imaginary roots).

If the coefficients are integers, \(\sqrt{\Delta}\) is rational—and the roots rational—exactly when \(\Delta\) is a perfect square.

3. Vieta’s formulae (roots ↔ coefficients)

For a monic equation the symmetric functions of the roots are read straight off the coefficients, with alternating signs:

Quadratic \(x^{2}+bx+c\): sum \(=-b\), product \(=c\).
Cubic \(x^{3}+bx^{2}+cx+d\) with roots \(\alpha,\beta,\gamma\): \(\alpha+\beta+\gamma=-b\), \(\alpha\beta+\beta\gamma+\gamma\alpha=c\), \(\alpha\beta\gamma=-d\).
General degree \(n\): the sum of the roots is \(-\dfrac{a_{n-1}}{a_{n}}\), the sum of products taken two at a time is \(\dfrac{a_{n-2}}{a_{n}}\), and so on; the product of all roots is \((-1)^{n}\dfrac{a_{0}}{a_{n}}\).

A clean way to remember it: build the equation as \(x^{n}-(\text{sum})x^{n-1}+(\text{sum of pairs})x^{n-2}-\cdots=0\).

4. Forming and transforming equations

To form an equation with prescribed roots, compute the symmetric functions and slot them into the template above. To get an equation whose roots are a function of the old ones (each root increased by 2, doubled, squared, reciprocated, …), express the new sum and product in terms of the old sum and product—you never have to solve the original.

A handy identity: \(\alpha^{2}+\beta^{2}=(\alpha+\beta)^{2}-2\alpha\beta\); more generally \(\sum\alpha_i^{2}=\left(\sum\alpha_i\right)^{2}-2\sum_{i\lt j}\alpha_i\alpha_j\).
Reciprocal roots: the equation with roots \(1/\alpha_i\) is obtained by reversing the coefficient list (replace \(x\) by \(1/x\) and clear denominators).

5. Fundamental Theorem of Algebra & multiplicity

Every polynomial equation of degree \(n\ge 1\) has at least one complex root; consequently it has exactly \(n\) roots in \(\mathbb{C}\), counted with multiplicity. If \((x-a)^{k}\) divides \(P(x)\) but \((x-a)^{k+1}\) does not, then \(a\) is a root of multiplicity \(k\); a root of multiplicity 1 is a simple root.

6. Conjugate-root theorems

Complex Conjugate Root Theorem. If a polynomial has real coefficients and \(\alpha+i\beta\) (\(\beta\neq 0\)) is a root, then \(\alpha-i\beta\) is also a root. So non-real roots arrive in conjugate pairs, and an odd-degree real polynomial must have at least one real root.
Surd Conjugate Root Theorem. If a polynomial has rational coefficients and \(p+\sqrt{q}\) (with \(\sqrt{q}\) irrational) is a root, then \(p-\sqrt{q}\) is also a root. The matching factor you divide out is \(\left(x-(p+\sqrt q)\right)\left(x-(p-\sqrt q)\right)=x^{2}-2px+(p^{2}-q)\).

Trap: “irrational roots always occur in pairs” is false in general. The pairing needs rational coefficients and a surd of the form \(p+\sqrt q\). For example \(x^{3}-2=0\) has a single irrational root \(\sqrt[3]{2}\); its other two roots are imaginary.

7. Rational Root Theorem

For an integer-coefficient polynomial \(a_{n}x^{n}+\cdots+a_{0}\) (\(a_{n}\neq 0,\ a_{0}\neq 0\)), any rational root \(\dfrac{p}{q}\) in lowest terms has \(p\mid a_{0}\) and \(q\mid a_{n}\). For a monic integer polynomial \(q=\pm 1\), so any rational root is an integer dividing \(a_{0}\). The theorem only produces a candidate list—each candidate must still be tested.

8. Special solving techniques

Even powers only. If only even powers appear (degree \(2n\)), substitute \(y=x^{2}\) to get a degree-\(n\) equation; each \(y\)-root gives \(x=\pm\sqrt{y}\).
Sum of all coefficients \(=0\). Then \(P(1)=0\), so \(x=1\) is a root and \((x-1)\) a factor.
Sum of even-power coefficients \(=\) sum of odd-power coefficients. Then \(P(-1)=0\), so \(x=-1\) is a root and \((x+1)\) a factor.
Roots in progression. If the roots are in A.P. write them \(a-d,\,a,\,a+d\); in G.P. write them \(\dfrac{a}{r},\,a,\,ar\); in H.P. work with reciprocals in A.P. Apply Vieta’s formulae to pin down \(a,d\) (or \(a,r\)).
Partly factored quartics of the form \((x-a)(x-b)(x-c)(x-d)=k\) with a hidden pairing can be reduced to a quadratic in a clever substitution such as \(y=x^{2}+(\text{linear})\).
Non-polynomial equations (with \(\sqrt{x}\) or fractional powers) can be turned into polynomial ones by a substitution like \(y=\sqrt{x}\); afterwards discard spurious roots that violate the original domain (e.g. \(\sqrt{x}\ge 0\)).

9. Reciprocal equations

A reciprocal equation reproduces itself when \(x\) is replaced by \(1/x\). Two types:

Type I: coefficients read the same forwards and backwards (\(a_{k}=a_{n-k}\)).
Type II: coefficients are equal in size but opposite in sign (\(a_{k}=-a_{n-k}\)).

Useful consequences: an odd-degree Type I equation has \(x=-1\) as a root; an odd-degree Type II equation has \(x=1\) as a root; an even-degree Type II equation has both \(x=1\) and \(x=-1\) as roots. For even degree, the substitution \(y=x+\dfrac{1}{x}\) (or \(y=x-\dfrac{1}{x}\)) halves the degree. A reciprocal equation never has \(0\) as a root.

10. Descartes’ Rule of Signs

Let \(s\) be the number of sign changes in the ordered coefficients of \(P(x)\) and \(p\) the number of positive roots. Then \(s-p\) is a non-negative even integer—so the positive roots number \(s,\ s-2,\ s-4,\ \ldots\). Counting sign changes in \(P(-x)\) bounds the negative roots the same way. The rule gives only upper bounds; what is left over after the maximum real roots are accounted for must be imaginary, which is how you derive a lower bound on imaginary roots.

11. Common exam traps

“\(n\) roots” means \(n\) in \(\mathbb{C}\) with multiplicity—not \(n\) distinct, nor \(n\) real.
Conjugate pairing applies for complex roots only with real coefficients, and for surd roots only with rational coefficients. Mixing these up loses marks.
The Rational Root Theorem lists candidates, not answers; always verify by substitution.
In Descartes’ Rule, “at most \(k\) positive roots” never means “exactly \(k\)”—subtract in steps of two.
When a substitution like \(y=x^{2}\) or \(y=\sqrt{x}\) is used, translate back carefully and reject roots outside the original domain.

Theory of Equations — Study Notes