Week 4: Algebra of Polynomials, Division & Roots

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2026-09-12T00:00:00.000Z

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Polynomial operations, long division, remainder and factor theorems, root multiplicity, turning points, and end behavior. # Mathematics I — Week 4: Algebra of Polynomials, Division & Roots IITM’s Week 4 covers **algebra of polynomials, addition/subtraction/multiplication/division, polynomial algorithms, graphs of po...

Mathematics I — Week 4: Algebra of Polynomials, Division & Roots

IITM’s Week 4 covers algebra of polynomials, addition/subtraction/multiplication/division, polynomial algorithms, graphs of polynomials, $x$ -intercepts, multiplicities, and end behavior.

TL;DR

A polynomial is built from powers of

x

with nonnegative integer exponents:

P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0

with:

a_n\neq0.

The important Week 4 ideas are:

degree and coefficients
polynomial addition/subtraction/multiplication
long division / synthetic-division style reasoning
factor theorem
remainder theorem
roots / zeros / $x$ -intercepts
multiplicity
how multiplicity affects the graph
leading coefficient
end behavior from degree + sign of leading coefficient

1. What is a polynomial?

A polynomial in

x

has the form:

\boxed{ P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 }

where:

$n$ is a nonnegative integer
coefficients $a_i$ are usually real numbers
$a_n\neq0$

Example:

P(x)=3x^4-2x^2+7x-5.

This is a polynomial of degree:

4.

2. What is not a polynomial?

These are not polynomials:

\frac1x

because:

\frac1x=x^{-1}

has a negative exponent.

Also:

\sqrt{x}=x^{1/2}

has a fractional exponent.

And:

2^x

has the variable in the exponent.

A polynomial requires exponents:

0,1,2,3,\dots

only.

3. Terms

Consider:

P(x)=4x^3-7x+2.

The terms are:

4x^3,\qquad -7x,\qquad 2.

Each term is typically:

\text{coefficient}\times x^{\text{power}}.

4. Coefficients

For:

P(x)=5x^4-3x^2+x-8,

the coefficients are:

5,\quad0,\quad-3,\quad1,\quad-8.

Notice the missing

x^3

term has coefficient:

0.

It is often useful to write:

P(x)=5x^4+0x^3-3x^2+x-8.

That becomes important during polynomial division.

5. Degree

The degree is the highest exponent with a nonzero coefficient.

Example:

P(x)=7x^5-2x^3+1.

Then:

\boxed{\deg P=5}

6. Leading term and leading coefficient

For:

P(x)=3x^6-8x^2+4,

the leading term is:

3x^6

and the leading coefficient is:

3.

These become extremely important for end behavior.

7. Constant polynomial

A nonzero constant:

P(x)=5

has degree:

0.

The zero polynomial:

P(x)=0

is special; its degree is usually treated as undefined.

8. Evaluating a polynomial

Given:

P(x)=2x^3-x+4,

find:

P(2).

Substitute:

P(2)=2(2)^3-2+4.

So:

=16-2+4

=18.

This is basic, but it powers the remainder theorem later.

9. Equality of polynomials

Two polynomials are equal if corresponding coefficients match.

Suppose:

ax^2+bx+c=3x^2-4x+7.

Then:

a=3,\qquad b=-4,\qquad c=7.

This is coefficient comparison.

10. Addition of polynomials

Add like powers.

Example:

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

and:

Q(x)=x^2-5x+4.

Then:

P(x)+Q(x)

equals:

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

so:

\boxed{ 4x^2-3x+5 }

11. Subtraction

Take care with signs.

P(x)-Q(x)

for:

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

and:

Q(x)=x^2-5x+4

gives:

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

Distribute the minus:

3x^2+2x+1-x^2+5x-4.

Therefore:

\boxed{ 2x^2+7x-3 }

12. Degree under addition

Usually:

\deg(P+Q)\leq\max(\deg P,\deg Q).

It can become smaller because leading terms may cancel.

Example:

P(x)=x^3+2

Q(x)=-x^3+x.

Then:

P+Q=x+2.

Although each original polynomial had degree

3

, their sum has degree

1

13. Multiplication of polynomials

Multiply every term by every term.

Example:

(x+2)(x+3)

gives:

x^2+3x+2x+6

so:

\boxed{ x^2+5x+6 }

14. Multiplying a binomial and polynomial

Example:

(x-2)(x^2+3x+4).

Distribute:

x(x^2+3x+4)-2(x^2+3x+4)

so:

x^3+3x^2+4x-2x^2-6x-8.

Combine:

\boxed{ x^3+x^2-2x-8 }

15. Degree under multiplication

For nonzero polynomials:

\boxed{ \deg(PQ)=\deg P+\deg Q }

Example:

If:

\deg P=3

and:

\deg Q=4,

then:

\deg(PQ)=7.

This is reliable because the leading terms multiply and cannot cancel.

16. Important algebraic identities

These save time constantly.

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

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

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

Also:

a^3-b^3=(a-b)(a^2+ab+b^2)

and:

a^3+b^3=(a+b)(a^2-ab+b^2).

These appear frequently during factorization.

17. Polynomial division

Polynomial division mirrors ordinary integer division.

For integers:

17=5(3)+2.

For polynomials:

P(x)=D(x)Q(x)+R(x).

where:

$P(x)$ = dividend
$D(x)$ = divisor
$Q(x)$ = quotient
$R(x)$ = remainder

and:

\deg R<\deg D.

This is the division algorithm for polynomials.

18. Polynomial division algorithm

Formally:

For polynomials

P(x)

and nonzero

D(x)

\boxed{ P(x)=D(x)Q(x)+R(x) }

where:

R(x)=0

or:

\deg R<\deg D.

This is a major structural result.

19. Example: simple polynomial division

Divide:

x^2+5x+6

by:

x+2.

We already know:

x^2+5x+6=(x+2)(x+3).

Therefore:

Q(x)=x+3

and:

R(x)=0.

20. Long division example

Divide:

x^3-2x^2+4x-8

by:

x-2.

First:

\frac{x^3}{x}=x^2.

Multiply:

x^2(x-2)=x^3-2x^2.

Subtract:

(x^3-2x^2+4x-8)-(x^3-2x^2)

leaving:

4x-8.

Now:

\frac{4x}{x}=4.

Multiply:

4(x-2)=4x-8.

Subtract:

0.

So:

\boxed{ x^3-2x^2+4x-8=(x-2)(x^2+4) }

21. Why missing terms matter in division

Suppose:

x^3+2x+1

is divided by something.

Write it as:

x^3+0x^2+2x+1.

The zero placeholder prevents coefficient misalignment.

This is especially useful for synthetic-division style calculations.

22. Remainder theorem

When a polynomial

P(x)

is divided by:

x-a,

the remainder is:

\boxed{ P(a) }

This is one of the highest-value results in Week 4.

23. Why remainder theorem works

Polynomial division gives:

P(x)=(x-a)Q(x)+R.

Because the divisor has degree

1

, the remainder must be constant.

Set:

x=a.

Then:

P(a)=(a-a)Q(a)+R

so:

P(a)=R.

Thus:

\boxed{R=P(a)}

24. Example: remainder theorem

Find the remainder when:

P(x)=x^3-4x+7

is divided by:

x-2.

No long division needed.

Compute:

P(2)=2^3-4(2)+7.

So:

8-8+7=7.

Therefore remainder:

\boxed{7}

Sign trap in the divisor

If the divisor is:

x+3,

rewrite it as:

x-(-3).

Therefore use:

P(-3),

not:

P(3).

Very common mistake.

26. Factor theorem

The factor theorem is a special case of the remainder theorem.

\boxed{ x-a\text{ is a factor of }P(x) \iff P(a)=0 }

Why?

If:

P(a)=0,

the remainder on division by

x-a

is zero.

Therefore:

x-a

divides

P(x)

exactly.

27. Example: factor theorem

Let:

P(x)=x^3-6x^2+11x-6.

Check whether:

x-1

is a factor.

Calculate:

P(1)=1-6+11-6=0.

Therefore:

x-1

is a factor.

Similarly:

P(2)=8-24+22-6=0.

So:

x-2

is also a factor.

And:

P(3)=27-54+33-6=0.

Thus:

P(x)=(x-1)(x-2)(x-3).

28. Root, zero and factor are connected

These three statements mean essentially the same thing:

P(a)=0

a\text{ is a root/zero}

x-a\text{ is a factor}.

And graphically:

(a,0)

is an

x

-intercept.

So:

\boxed{ \text{root} \leftrightarrow \text{factor} \leftrightarrow x\text{-intercept} }

This is one of the central conceptual connections of Week 4.

29. Fundamental structure of factored polynomials

If:

P(x)=a(x-r_1)(x-r_2)\cdots(x-r_n),

then each:

r_i

is a root.

The graph intersects or touches the

x

-axis at those values.

30. Multiplicity

Suppose:

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

Then:

x=2

is a root of multiplicity:

3.

And:

x=-1

is a root of multiplicity:

2.

Multiplicity means:

how many times the same factor occurs.

31. Example

P(x)=(x-4)^5.

Then root:

x=4

has multiplicity:

5.

32. Total multiplicity and degree

For a polynomial fully factored over a suitable number system, the sum of root multiplicities equals the polynomial's degree.

Example:

P(x)=(x-1)^2(x+3)^4.

Total degree:

2+4=6.

33. Odd multiplicity

If a root has odd multiplicity, the graph typically crosses the

x

-axis.

Example:

y=(x-1)^3.

At:

x=1,

the graph crosses through the axis.

For multiplicity

3,5,\dots

, it often flattens more near the crossing.

34. Even multiplicity

If a root has even multiplicity, the graph touches the

x

-axis and turns around.

Example:

y=(x-2)^2.

At:

x=2,

the graph touches but does not cross.

Mental shortcut:

\boxed{ \begin{aligned} \text{odd multiplicity} &\Rightarrow \text{cross}\\ \text{even multiplicity} &\Rightarrow \text{touch/bounce} \end{aligned} }

35. Why odd/even multiplicity behaves differently

Near a root

r

, consider:

(x-r)^m.

m

is odd, changing from

x<r

x>r

changes the sign.

m

is even, the sign remains nonnegative locally.

So the graph:

crosses if sign changes
touches if sign does not change

This is a deeper reason, not just a memorized graph rule.

36. Example: graph behavior from factors

Consider:

P(x)=(x+2)(x-1)^2(x-4)^3.

Roots:

x=-2,\quad1,\quad4.

Multiplicities:

$-2$ : multiplicity $1$ → crosses
$1$ : multiplicity $2$ → touches
$4$ : multiplicity $3$ → crosses, with flattening

This gives a huge amount of graph information before plotting anything.

37. Degree of that example

For:

P(x)=(x+2)(x-1)^2(x-4)^3,

degree:

1+2+3=6.

So it is an even-degree polynomial.

38. Leading coefficient from factored form

If:

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

the leading coefficient is:

2.

Why?

The highest-degree terms multiply:

2(x^2)(x^3)=2x^5.

So degree:

5

and leading coefficient:

2.

39. End behavior

End behavior asks:

What happens to $P(x)$ as $x\to+\infty$ and $x\to-\infty$ ?

The leading term controls this.

If:

P(x)=a_nx^n+\text{lower-degree terms},

then for very large

|x|

P(x)\approx a_nx^n.

So only two things matter:

parity of degree $n$
sign of leading coefficient $a_n$

40. Why lower-degree terms stop mattering

Consider:

P(x)=5x^6-1000x^2+4x-9.

For huge

x

x^6

grows much faster than:

x^2,\quad x,\quad1.

Therefore:

5x^6

dominates.

This is the dominant-term principle.

41. Even degree + positive leading coefficient

Example:

P(x)=x^4.

As:

x\to+\infty,

P(x)\to+\infty.

As:

x\to-\infty,

P(x)\to+\infty.

So both ends go up:


text
↑             ↑
 \           /
  \_________/

Rule:

\boxed{ n\text{ even},\ a_n>0 \Rightarrow \text{up/up} }

42. Even degree + negative leading coefficient

Example:

P(x)=-x^4.

Then both ends go downward:

x\to\pm\infty \Rightarrow P(x)\to-\infty.

Rule:

\boxed{ n\text{ even},\ a_n<0 \Rightarrow \text{down/down} }

43. Odd degree + positive leading coefficient

Example:

P(x)=x^3.

Then:

x\to-\infty \Rightarrow P(x)\to-\infty

and:

x\to+\infty \Rightarrow P(x)\to+\infty.

So:


text
          /
        /
------/
    /
  /

Rule:

\boxed{ n\text{ odd},\ a_n>0 \Rightarrow \text{left down, right up} }

44. Odd degree + negative leading coefficient

Example:

P(x)=-x^3.

Then:

x\to-\infty \Rightarrow P(x)\to+\infty

and:

x\to+\infty \Rightarrow P(x)\to-\infty.

Rule:

\boxed{ n\text{ odd},\ a_n<0 \Rightarrow \text{left up, right down} }

45. End-behavior master table

Degree	Leading coefficient	Left end	Right end
even	positive	up	up
even	negative	down	down
odd	positive	down	up
odd	negative	up	down

This is one of the most exam-useful tables in Week 4.

46. Compact intuition

Even degree

Both ends move in the same direction.

Odd degree

The ends move in opposite directions.

Then the leading coefficient tells you the right-end direction:

positive → right end up
negative → right end down

That is often easier than memorizing four separate cases.

47. Example: end behavior

Consider:

P(x)=-3x^7+4x^2-8.

Degree:

7

odd.

Leading coefficient:

-3<0.

Therefore:

x\to-\infty \Rightarrow P(x)\to+\infty

and:

x\to+\infty \Rightarrow P(x)\to-\infty.

No graphing required.

48. Graphing a polynomial systematically

Consider:

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

A useful workflow:

Step 1: degree

2+3=5

Odd degree.

Step 2: leading coefficient

Positive.

So:

\text{left down, right up}.

Step 3: roots

x=-1,\quad2.

Step 4: multiplicities

At:

x=-1,

multiplicity

2

→ touches.

At:

x=2,

multiplicity

3

→ crosses.

Step 5: $y$ -intercept

Set:

x=0.

Then:

P(0)=(1)^2(-2)^3=-8.

So the graph passes through:

(0,-8).

You can now sketch the rough graph with surprisingly little computation.

49. Maximum number of roots

A polynomial of degree

n

can have at most:

\boxed{n}

real roots.

For example, a cubic can have at most

3

real roots.

A degree-

5

polynomial can have at most

5

real roots.

50. Maximum number of turning points

A degree-

n

polynomial can have at most:

\boxed{n-1}

turning points.

Examples:

degree $1$ : at most $0$
degree $2$ : at most $1$
degree $3$ : at most $2$
degree $4$ : at most $3$

This gives useful graph sanity checks.

51. Important distinction: degree vs number of roots

A degree-

4

polynomial does not need to have four distinct real roots.

Examples:

x^4+1

has no real roots.

(x-1)^4

has only one distinct real root:

x=1

but with multiplicity

4

So:

degree gives the maximum possible number of distinct real roots, not a guarantee.

52. Polynomial sign analysis

Factored form can also tell where the function is positive or negative.

Example:

P(x)=(x-1)(x-3).

Roots divide the real line:

(-\infty,1),\quad(1,3),\quad(3,\infty).

Test one value from each interval.

$x=0$

(-1)(-3)>0.

$x=2$

(1)(-1)<0.

$x=4$

(3)(1)>0.

Therefore:

P(x)>0

for:

x<1

or:

x>3.

53. Multiplicity gives a shortcut for sign changes

At an odd-multiplicity root, sign changes.

At an even-multiplicity root, sign stays the same.

So when moving across roots from left to right:

odd multiplicity → flip sign
even multiplicity → preserve sign

Very useful for complex factorizations.

54. Example

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

At:

x=-2,

multiplicity

2

→ no sign change.

At:

x=1,

multiplicity

3

→ sign changes.

This predicts graph crossing behavior and sign intervals simultaneously.

55. Remainder theorem with parameters

Suppose:

P(x)=x^3+kx+2

leaves remainder

6

when divided by:

x-1.

By remainder theorem:

P(1)=6.

Thus:

1+k+2=6

so:

k=3.

This kind of question is very common.

56. Factor theorem with parameters

Suppose:

P(x)=x^3+kx^2-4x-4

and

x-2

is a factor.

Then:

P(2)=0.

So:

8+4k-8-4=0.

Thus:

4k-4=0

and:

\boxed{k=1}

Again: much faster than doing full division.

57. Finding remaining factors

Suppose:

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

and you know:

x-1

is a factor.

Divide by

x-1

P(x)=(x-1)(x^2-5x+6).

Then factor:

x^2-5x+6=(x-2)(x-3).

Therefore:

\boxed{ P(x)=(x-1)(x-2)(x-3) }

58. Synthetic-division intuition

When dividing by:

x-a,

full polynomial long division can be compressed into operations on coefficients.

For:

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

divide by:

x-1.

Use coefficients:

1,\ -6,\ 11,\ -6.

With:

a=1.

Procedure conceptually:


text
1    -6     11    -6
     ↓
bring 1 down
multiply by 1
add
repeat

The final coefficients become:

1,\ -5,\ 6

with remainder:

0.

So quotient:

x^2-5x+6.

Even if IITM presents this differently, the underlying idea is just efficient division by a linear factor.

59. Polynomial interpolation intuition

A subtle but useful idea:

A degree-

n

polynomial is essentially determined by

n+1

appropriate data points.

For example:

degree $1$ line → $2$ points
degree $2$ quadratic → $3$ points
degree $3$ cubic → $4$ points

This builds on Week 2 and Week 3.

You do not need full interpolation machinery yet, but the pattern is useful.

60. How Week 3 generalizes into Week 4

For quadratics:

ax^2+bx+c.

Week 3 studied:

roots
vertex
intercepts
direction of opening

Week 4 says:

What if degree is $3,4,5,\dots$ ?

Then:

roots still matter
factors still matter
intercepts still matter
degree becomes central
multiplicity becomes richer
end behavior generalizes the "opens up/down" idea

So Week 4 is really quadratic graph reasoning generalized to all polynomials.

Common mistake: degree after simplification

Suppose:

P(x)=x^4-x^4+3x^2.

You cannot say degree

4

just because

x^4

appears before simplification.

Simplify:

P(x)=3x^2.

Therefore:

\boxed{\deg P=2}

Common mistake: root sign

If a factor is:

x+5,

the root is:

x=-5,

not:

5.

Because:

x+5=0 \Rightarrow x=-5.

Common mistake: multiplicity

If:

P(x)=(x-2)^4(x+3),

there are not five distinct roots.

There are two distinct roots:

2,\quad -3.

with multiplicities:

4,\quad1.

Common mistake: even-degree graphs

"Even degree" does not automatically mean both ends up.

You must also check the leading coefficient.

Example:

-x^4

has both ends down.

Common mistake: odd-degree graphs

Similarly, odd degree does not always mean left down/right up.

That is only when the leading coefficient is positive.

For:

-x^3,

the orientation reverses.

Common mistake: factors vs terms

In:

(x-2)(x+5),

these are factors.

In:

x^2+3x-10,

the pieces:

x^2,\quad3x,\quad-10

are terms.

Do not mix the vocabulary.

Common mistake: remainder degree

If dividing by a quadratic:

D(x)

with degree

2

, the remainder may be linear:

ax+b.

It does not have to be constant.

General rule:

\deg R<\deg D.

The remainder theorem gives a constant specifically because:

x-a

has degree

1

68. High-value formula/concept sheet

General polynomial

\boxed{ P(x)=a_nx^n+\cdots+a_0 }

Division algorithm

\boxed{ P(x)=D(x)Q(x)+R(x) }

with:

\deg R<\deg D.

Remainder theorem

Dividing by

x-a

\boxed{ R=P(a) }

Factor theorem

\boxed{ x-a\mid P(x) \iff P(a)=0 }

Root-factor-intercept link

\boxed{ P(a)=0 \iff a\text{ root} \iff x-a\text{ factor} \iff (a,0)\text{ x-intercept} }

End behavior

Determined by:

\boxed{ \text{degree parity + sign of leading coefficient} }

Multiplicity

\boxed{ \text{odd}\Rightarrow\text{cross} }

\boxed{ \text{even}\Rightarrow\text{touch} }

69. Exam workflow for a graph question

Given a polynomial:

P(x),

I would inspect in this order:


text
1. Simplify / factor if possible
2. Find degree
3. Find leading coefficient
4. Predict end behavior
5. Find roots
6. Note each multiplicity
7. Determine cross/touch behavior
8. Find y-intercept P(0)
9. Check overall sign/shape
10. Sanity-check turning points

This is much more reliable than trying to sketch directly.

70. What you should actually master

By the end of Week 4, you should be able to:

Identify whether an expression is a polynomial.
Find degree, leading term and leading coefficient.
Add/subtract polynomials.
Multiply polynomials.
Perform or understand polynomial division.
Use: $P=DQ+R.$
Use the remainder theorem.
Use the factor theorem.
Convert between roots and factors.
Find $x$ -intercepts from factors.
Identify root multiplicity.
Predict cross/touch behavior.
Infer end behavior from degree and leading coefficient.
Use factored form to sketch a rough polynomial graph.
Solve parameter questions using $P(a)=0$ or $P(a)=R$ .

If those are comfortable, Week 4 is under control.

71. Weeks 1–4 conceptual progression


text
WEEK 1
Sets → Relations → Functions
                 ↓
WEEK 2
Coordinates → Straight Lines
                 ↓
WEEK 3
Quadratic Functions
                 ↓
WEEK 4
General Polynomial Functions

The course is gradually widening the class of functions you can understand geometrically and algebraically.

Quick Recall & Flashcards

Test your active recall before proceeding to practice problems. Format:

Question :: Answer.

Remainder Theorem for $P(x)$ divided by $x - c$ :: Remainder is $P(c)$
Factor Theorem condition :: $x - c$ is a factor of $P(x)$ if and only if $P(c) = 0$
Behavior of graph at root with even multiplicity :: Touches x-axis and turns around (tangent)
Behavior of graph at root with odd multiplicity :: Crosses the x-axis
Maximum number of turning points for degree $n$ polynomial :: $n - 1$
End behavior of polynomial is determined by :: The leading term $a_n x^n$

Practice & Self-Check

Attempt these diagnostic questions to verify your operational understanding.

Question 1

What is the remainder when the polynomial

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

is divided by

x - 2

NAT

3 Unit Assessment

Question 2

If a polynomial has a root at

x = 3

with multiplicity 2, what does the graph do at

x = 3

MCQ

3 Unit Assessment

A. Crosses the x-axis directly

B. Touches (is tangent to) the x-axis and turns around without crossing

C. Has a vertical asymptote

D. Has a jump discontinuity

What this note is really saying

Mathematics I — Week 4: Algebra of Polynomials, Division & Roots

TL;DR

1. What is a polynomial?

2. What is not a polynomial?

3. Terms

4. Coefficients

5. Degree

6. Leading term and leading coefficient

7. Constant polynomial

8. Evaluating a polynomial

9. Equality of polynomials

10. Addition of polynomials

11. Subtraction

12. Degree under addition

13. Multiplication of polynomials

14. Multiplying a binomial and polynomial

15. Degree under multiplication

16. Important algebraic identities

17. Polynomial division

18. Polynomial division algorithm

19. Example: simple polynomial division

20. Long division example

21. Why missing terms matter in division

22. Remainder theorem

23. Why remainder theorem works

24. Example: remainder theorem

26. Factor theorem

27. Example: factor theorem

28. Root, zero and factor are connected

29. Fundamental structure of factored polynomials

30. Multiplicity

31. Example

32. Total multiplicity and degree

33. Odd multiplicity

34. Even multiplicity

35. Why odd/even multiplicity behaves differently

36. Example: graph behavior from factors

37. Degree of that example

38. Leading coefficient from factored form

39. End behavior

40. Why lower-degree terms stop mattering

41. Even degree + positive leading coefficient

42. Even degree + negative leading coefficient

43. Odd degree + positive leading coefficient

44. Odd degree + negative leading coefficient

45. End-behavior master table

46. Compact intuition

Even degree

Odd degree

47. Example: end behavior

48. Graphing a polynomial systematically

Step 1: degree

Step 2: leading coefficient

Step 3: roots

Step 4: multiplicities

Step 5: yyy-intercept

49. Maximum number of roots

50. Maximum number of turning points

51. Important distinction: degree vs number of roots

52. Polynomial sign analysis

x=0x=0x=0

x=2x=2x=2

x=4x=4x=4

53. Multiplicity gives a shortcut for sign changes

54. Example

55. Remainder theorem with parameters

56. Factor theorem with parameters

57. Finding remaining factors

58. Synthetic-division intuition

59. Polynomial interpolation intuition

60. How Week 3 generalizes into Week 4

68. High-value formula/concept sheet

General polynomial

Division algorithm

Remainder theorem

Factor theorem

Root-factor-intercept link

End behavior

Multiplicity

Step 5: $y$ -intercept

$x=0$

$x=2$

$x=4$