Polynomials — SPI Mathematics Formulas and Traps

Degree, Value and Zeroes of a Polynomial

Value of a polynomial and what a zero is

Zero of a polynomial

p(a) = 0 \qquad \text{zero of } ax + b = -\frac{b}{a}

Degree of a polynomial and the name for each degree

Name	Degree	Example
Constant polynomial	0	7
Linear polynomial	1	3x + 2
Quadratic polynomial	2	x² − 4x + 1
Cubic polynomial	3	2x³ + x − 5
Biquadratic (quartic) polynomial	4	x⁴ − 5x² + 6 A biquadratic is a quadratic in x², so its degree is 4, not 2.

The same names are used for equations: a cubic equation has degree 3.

Reading the zeroes of a polynomial from its graph

What the graph does	Number of distinct zeroes
A straight line crosses the x-axis once	1
A parabola cuts the x-axis at two points	2
A parabola only touches the x-axis at one point	1 Touching counts: the point where it touches is a zero.
A parabola never meets the x-axis	0
A cubic curve crosses the x-axis three times	3

Count the meeting points with the x-axis, not with the y-axis.

Common traps

Taking the degree from the first term

The degree is the highest power anywhere in the polynomial.

2x - x^3 + 4

has degree 3, even though it starts with an x term.

Treating a root sign as a power of x

\sqrt{2}

is a constant and does not change the degree.

\sqrt{x}

is

x^{1/2}

, which makes the expression not a polynomial.

Squaring after attaching the minus sign

At

x = -2

, the term

-3x^2

is

-3 \times 4 = -12

, not

+12

. Square the number first, then multiply by the coefficient with its sign.

Counting where the graph meets the y-axis

Zeroes are where the graph meets the x-axis. A curve that cuts the y-axis at 3 has

p(0) = 3

, which says nothing about its zeroes.

Missing a zero where the curve only touches

A parabola that just touches the x-axis and turns back still meets it there, so it has exactly one distinct zero, not none.

Factorising Polynomials

Learn this subtopic in the notes

Factorising a quadratic polynomial by splitting the middle term

Product of two brackets

(x + p)(x + q) = x^2 + (p + q)x + pq

Factorising by grouping and by substitution

Grouping pattern

ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)

Common traps

Checking only the product

(x + 6)(x - 4)

and

(x - 6)(x + 4)

both give the constant

-24

. The middle term decides: the first gives

+2x

, the second

-2x

.

Comparing terms of different powers

Match the coefficient of x with the coefficient of x, and the constant with the constant. In

x^2 + kx + 8 = (x + 2)(x + 4)

, k is 6, the x coefficient, not 8.

Taking out a positive factor from a negative pair

In

x^3 - 2x^2 - 5x + 10

, take

-5

out of the second pair:

x^2(x - 2) - 5(x - 2) = (x - 2)(x^2 - 5)

. Taking out

+5

leaves

(-x + 2)

, which does not match.

Leaving the answer in t

After factorising in t, replace t by the bracket it stood for.

(t - 3)(3t + 2)

with

t = a + b

becomes

(a + b - 3)(3a + 3b + 2)

.

Division Algorithm, Remainder and Factor Theorems

Learn this subtopic in the notes

The division algorithm for polynomials

Division algorithm

p(x) = g(x)\,q(x) + r(x), \quad \deg r < \deg g \qquad \deg q = \deg p - \deg g

The remainder theorem

Remainder theorem

\text{remainder of } p(x) \div (x - a) = p(a)

The factor theorem

Factor theorem

(x - a) \text{ is a factor of } p(x) \text{ when } p(a) = 0

Common traps

Skipping a missing power

x^3 - 7x + 6

has no

x^2

term. Write it as

x^3 + 0x^2 - 7x + 6

before dividing, or the subtraction lines up the wrong columns.

Expecting a number as every remainder

The remainder only has to be of lower degree than the divisor. Divided by a quadratic, the remainder can be linear, such as

3x + 1

.

Putting in the wrong sign

For the divisor

x + 3

, put

x = -3

, not

x = 3

. The number you put in is the one that makes the divisor 0.

Mixing up the factor and the zero

The factor

x + 5

goes with the zero

-5

, and the factor

x - 5

with the zero 5. Test a factor by putting in the number that makes it 0.

Zeroes and Coefficients

Learn this subtopic in the notes

Sum and product of the zeroes of a quadratic polynomial

Zeroes of a quadratic polynomial

\alpha + \beta = -\frac{b}{a} \qquad \alpha\beta = \frac{c}{a} \qquad \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta

Forming a quadratic polynomial from its zeroes

Quadratic from its zeroes

k\left[x^2 - (\alpha + \beta)x + \alpha\beta\right]

Zeroes and coefficients of a cubic polynomial

Zeroes of a cubic polynomial

\alpha + \beta + \gamma = -\frac{b}{a} \qquad \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \qquad \alpha\beta\gamma = -\frac{d}{a}

Common traps

Dropping the minus in the sum

For

2x^2 + 3x + 4

, the sum of the zeroes is

-\frac{3}{2}

and the product 2, so

\frac{1}{\alpha} + \frac{1}{\beta} = -\frac{3}{4}

, not

\frac{3}{4}

.

Reading signs from the product alone

A positive product says only that the zeroes share a sign. Look at the sum to tell whether both are positive or both are negative.

Thinking only one polynomial fits

x^2 - 5x + 4

,

2x^2 - 10x + 8

and

-x^2 + 5x - 4

all have the zeroes 1 and 4. An option that is a multiple of your answer is still right.

Adding instead of subtracting in a conjugate product

(3 + \sqrt{2})(3 - \sqrt{2}) = 9 - 2 = 7

, not

9 + 2

. The square of the root part is always subtracted.

Using the quadratic sign for a cubic product

For a quadratic the product of the zeroes is

+\frac{c}{a}

, but for a cubic it is

-\frac{d}{a}

. The product of the zeroes of

x^3 - 2x + 5

is

-5

.

Polynomials formulas

Degree, Value and Zeroes of a Polynomial

Value of a polynomial and what a zero is

Degree of a polynomial and the name for each degree

Reading the zeroes of a polynomial from its graph

Common traps

Factorising Polynomials

Factorising a quadratic polynomial by splitting the middle term

Factorising by grouping and by substitution

Common traps

Division Algorithm, Remainder and Factor Theorems

The division algorithm for polynomials

The remainder theorem

The factor theorem

Common traps

Zeroes and Coefficients

Sum and product of the zeroes of a quadratic polynomial

Forming a quadratic polynomial from its zeroes

Zeroes and coefficients of a cubic polynomial

Common traps

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