713 words

4 min read

Visual companion

Python

Type and operator map

Python Week 1: the first filter for runtime behavior

Open gallery Related surface

View

Revision summary

What this note is really saying

Short form

# Mathematics I · Week 4 — Polynomials Beyond quadratics: degree, operations, factor theorem, multiplicity, and end behaviour. ## Week map Polynomial structure → degree and leading term → add/multiply/divide → remainder idea → factor theorem → zeros and multiplicity → sketch from factors.

Mathematics I · Week 4 — Polynomials

Beyond quadratics: degree, operations, factor theorem, multiplicity, and end behaviour.

Week map

Polynomial structure → degree and leading term → add/multiply/divide → remainder idea → factor theorem → zeros and multiplicity → sketch from factors.

Polynomial notation

$p(x) = a_n x^n + \cdots + a_1 x + a_0$ → polynomial in $x$ → coefficients real → $p(x) = 3x^3 - 2x + 7$ .
Degree $n$ → highest power with $a_n \neq 0$ → $5x^2 - 9$ has degree 2.
Leading term $a_n x^n$ → dominates for large $|x|$ → end behaviour driver.
Zero / root $r$ → $p(r) = 0$ → $(x-r)$ is a factor (over reals, if fully factorable).

Multiplicity

(x-r)^m

divides

p(x)

with largest

m

, root

r

has multiplicity

m

$m=1$ → graph crosses axis at $r$ .
$m=2$ → often touches and turns (bounce).
$m=3$ → crosses but flattens (flattened S).

Mini-example:

p(x) = (x-1)^2(x+3)

. Zeros:

1

(mult 2),

-3

(mult 1). Degree 3.

Operations

Add/subtract: combine like terms only ( $x^2$ with $x^2$ , not with $x$ ).
Multiply: distribute each term; exponents add: $x^a \cdot x^b = x^{a+b}$ .
Long division: divide $p(x)$ by $(x-a)$ ; remainder degree less than divisor; if remainder 0, $(x-a)$ is factor.

Mini-example:

(2x^2 + 3x - 5) + (x^2 - 3x + 1) = 3x^2 - 4

Multiply:

(x+2)(x^2 - x + 1) = x^3 + x^2 - x + 2

Factor theorem

(x - a)

is a factor of

p(x)

iff

p(a) = 0

Useful workflow: test small integers for rational roots when leading coefficient is 1 or small.

Mini-example:

p(x) = x^3 - 4x^2 + x + 6

. Try

a=2

8-16+2+6=0

→

(x-2)

factor. Divide to get

x^2 - 2x - 3 = (x-3)(x+1)

. Full:

(x-2)(x-3)(x+1)

End behaviour

For large

|x|

p(x) \approx a_n x^n

$n$	$a_n > 0$	$a_n < 0$
even	both ends up	both ends down
odd	left down, right up	left up, right down

Mini-example:

-2x^4 + \cdots

→ even degree 4, negative leading → both ends down.

Pattern families

Easy — Degree and evaluation

Combine polynomials; state degree and leading coefficient.
Evaluate $p(2)$ , $p(-1)$ by substitution.
Identify $y$ -intercept as $p(0) = a_0$ .

Medium — Factor using theorem

Test candidates; divide out linear factor; repeat.
Build polynomial from given zeros with multiplicity in factored form.
Expand factored form to standard when needed.

Hard — Graph reasoning

Sketch sign chart from factorization.
Relate multiplicity to touch/cross at each zero.
Parameter $k$ so $x-a$ divides $p(x)-k$ (horizontal shift of roots).

Worked mini-examples

Example 1 — Degree.

p(x) = 7x^4 - x^7 + 2x

. Rewrite standard:

-x^7 + 7x^4 + 2x

. Degree 7, leading coeff

-1

Example 2 — Factor test.

p(x) = x^3 - 6x^2 + 11x - 6

p(1)=0

p(2)=0

p(3)=0

→

(x-1)(x-2)(x-3)

Example 3 — Multiplicity sketch.

p(x) = x^2(x-4)^3

. Zeros:

0

mult 2 (touch),

4

mult 3 (cross flat). Degree 5. Ends: positive leading (implicit

x^5

) → left down, right up.

Example 4 — Remainder. Divide

x^3 + 2x - 5

(x-1)

: synthetic or long division gives remainder

-4

(same as

p(1)

Traps

Degree of zero polynomial handled specially; constant nonzero has degree 0.
Dropping terms when adding — write all powers or use columns.
Confusing zero of function with $y$ -intercept ( $x=0$ input).
Assuming all even-multiplicity roots “bounce” without checking local shape — still need sign chart.
Wrong end behaviour when only middle terms given — identify leading term first.

Diagnostic (try yourself)

What is the degree and leading coefficient of $p(x) = 4x - x^3 + 2x^5 - 7$ ?
If $p(x) = (x+2)^2(x-5)$ , list all real zeros and their multiplicities.
Use the factor theorem to decide whether $(x+1)$ is a factor of $p(x) = x^3 + 3x^2 - x - 3$ .
Multiply and simplify: $(x^2 - 3)(2x^2 + x - 4)$ .
Describe end behaviour (left and right) of $f(x) = -3x^6 + 2x^2 - 1$ as $x \to \pm\infty$ .