Polynomials — CDS Mathematics Formulas and Traps

Degree, Zeros and Coefficients

Degree, identities and integer values

Degree of a product

\deg(fg) = \deg f + \deg g

Zeros and coefficients

Cubic: sum of zeros

\alpha + \beta + \gamma = -\dfrac ba, \quad \alpha\beta + \beta\gamma + \gamma\alpha = \dfrac ca, \quad \alpha\beta\gamma = -\dfrac da

Common traps

Equal degrees can cancel

(x^3 + x) - (x^3 + 1) = x - 1

: the degree dropped from

3

to

1

. So the degree of a sum is only 'at most' the larger degree.

Divide by a

The relations use

\dfrac ba

,

\dfrac ca

,

\dfrac da

. For a non-monic cubic,

\sum\alpha^2 = \dfrac{b^2 - 2ac}{a^2}

, not

b^2 - 2c

.

The Remainder Theorem

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Remainder on division by a linear factor

Remainder theorem

f(x) \div (x - a) \;\Rightarrow\; \text{remainder} = f(a)

Remainder on division by a quadratic

Linear remainder

r(x) = \dfrac{(x - \beta)f(\alpha) - (x - \alpha)f(\beta)}{\alpha - \beta}

Divisibility of xⁿ ± aⁿ

Factor of xⁿ − aⁿ

x^n - a^n = (x - a)\left(x^{n - 1} + x^{n - 2}a + \cdots + a^{n - 1}\right)

Common traps

x + a means substitute −a

Dividing by

x + 3

leaves

f(-3)

, not

f(3)

. The wrong sign gives a remainder of the right size and wrong value, and it is usually printed.

The remainder is not a number

On division by a quadratic the remainder is usually a linear expression like

3 - x

. An option that is a plain number fits only if the two values happen to be equal.

Even n and a plus sign

x^n + y^n

with

n

even is never divisible by

x + y

: at

x = -y

it equals

2y^n

. Knowing

n

is even therefore answers the question — with a 'no'.

The Factor Theorem

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Testing for a factor

Factor theorem

(x - a) \mid f(x) \iff f(a) = 0

Finding unknown coefficients

Quadratic factor

(x - \alpha)(x - \beta) \mid f(x) \iff f(\alpha) = f(\beta) = 0

Common traps

A factor of one is not a factor of both

When asked what divides BOTH polynomials, test each candidate on each polynomial. A root of one alone is not enough.

Factorise the quadratic factor first

Substituting a quadratic like

x^2 - 4x + 3

is not possible directly; its roots

1

and

3

are what you substitute. Getting a root's sign wrong gives two consistent but wrong equations.

Factorisation of Polynomials

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Cubics: find a root, then divide

Rational root test (monic)

x^3 + bx^2 + cx + d:\ \text{integer roots divide } d

Pair the brackets and substitute

Pairing four brackets

x(x + 3)\cdot(x + 1)(x + 2) = t(t + 2), \quad t = x^2 + 3x

Factorising with identities

A useful quartic

a^4 + a^2b^2 + b^4 = (a^2 + ab + b^2)(a^2 - ab + b^2)

Common traps

Check the sign of each root

If

f(-2) = 0

, the factor is

x + 2

, not

x - 2

. Options are built from the same numbers with the signs changed, so verify one root in the original.

Pair by equal sums

In

x(x + 2)(x + 3)(x + 5)

, pair

0

with

5

and

2

with

3

(both sum to

5

). Pairing

x

with

x + 2

gives two different quadratics and no substitution.

Plus or minus in the linear factor

u^3 + v^3 + w^3 - 3uvw

has the factor

u + v + w

with every sign PLUS. An option with one term negated is not a factor, even though it looks close.

HCF and LCM of Polynomials

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Reading off the HCF and LCM

HCF and LCM

\text{HCF}: \text{common factors, lowest power}; \quad \text{LCM}: \text{all factors, highest power}

HCF × LCM = product

Product rule

p(x)\,q(x) = \text{HCF}(p, q)\times\text{LCM}(p, q)

Unknowns from a given HCF

Common root

k^2 + ak + b = 0,\ k^2 + ck + d = 0 \;\Rightarrow\; k = \dfrac{d - b}{a - c}

Common traps

Lowest power for the HCF

(x + 1)^3

and

(x + 1)^2(x - 1)

share

(x + 1)^2

, not

(x + 1)^3

. Taking the higher power belongs to the LCM.

Only for two polynomials

p q = \text{HCF}\times\text{LCM}

is true for a pair. For three polynomials the product is generally not HCF

\times

LCM.

x + k has root −k

If the HCF is

x + k

, substitute

x = -k

. The answer for

k

then comes out with the opposite sign to the root.

Polynomials formulas

Degree, Zeros and Coefficients

Degree, identities and integer values

Zeros and coefficients

Common traps

The Remainder Theorem

Remainder on division by a linear factor

Remainder on division by a quadratic

Divisibility of xⁿ ± aⁿ

Common traps

The Factor Theorem

Testing for a factor

Finding unknown coefficients

Common traps

Factorisation of Polynomials

Cubics: find a root, then divide

Pair the brackets and substitute

Factorising with identities

Common traps

HCF and LCM of Polynomials

Reading off the HCF and LCM

HCF × LCM = product

Unknowns from a given HCF

Common traps

More CDS Mathematics formula sheets