Polynomial study guide

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Writing Polynomials in Standard Form

Standard Form

Write terms in descending order of exponents.

Example:
y=2x3−6x+3x2−x4+12y = 2x^3 - 6x + 3x^2 - x^4 + 12y=2x3−6x+3x2−x4+12

Standard form:

y=−x4+2x3+3x2−6x+12y = -x^4 + 2x^3 + 3x^2 - 6x + 12y=−x4+2x3+3x2−6x+12

Classifying by Degree

End Behavior Rules

Look at the leading term only.

Factoring & Solving Polynomial Equations

Steps to Solve by Factoring

1. Set equation = 0

2. Factor completely

3. Set each factor equal to 0

4. Solve

Common Factoring Methods

• GCF
• Difference of Squares
• Trinomials
• Grouping

Example:
x2−11x=−24x^2 - 11x = -24x2−11x=−24

Move all terms:

x2−11x+24=0x^2 - 11x + 24 = 0x2−11x+24=0

Factor:

(x−3)(x−8)(x - 3)(x - 8)(x−3)(x−8)

Solutions: 3, 8

Zeros & Multiplicity

Zero

A value that makes the function = 0.

Multiplicity

How many times a zero repeats.

Example:

y=(x−2)3y = (x - 2)^3y=(x−2)3

Zero: 2
Multiplicity: 3

• Odd multiplicity → graph crosses x-axis
• Even multiplicity → graph touches and bounces

Graphing Polynomials

When graphing:

✔ Find zeros
✔ Determine multiplicity
✔ Check end behavior
✔ Look for relative max/min

Long Division of Polynomials

Used when divisor is more than one term.

Steps:

1. Divide

2. Multiply

3. Subtract

4. Bring down

5. Repeat

Synthetic Division

Shortcut when dividing by:

(x−c)(x - c)(x−c)

Steps:

1. Write coefficients

2. Use zero (c)

3. Multiply & add down

4. Last number = remainder

Remainder Theorem

If dividing by:

(x−a)(x - a)(x−a)

Then remainder = P(a)

Rational Root Theorem

Possible rational roots:

factors of constant. factors of leading coefficient\fraction. factors of leading coefficient.

Example:

P(x)=x3+4x2−10x+6P(x) = x^3 + 4x^2 - 10x + 6P(x)=x3+4x2−10x+6

Factors of 6: ±1, ±2, ±3, ±6
Factors of 1: ±1

Possible roots:
±1, ±2, ±3, ±6

Descartes’ Rule of Signs

Positive Zeros

Count sign changes in P(x)P(x)P(x)

Negative Zeros

Count sign changes in P(−x)P(-x)P(−x)

Possible answers decrease by 2.

Complex & Irrational Zeros Rules

If coefficients are rational:

• If a+bia + bia+bi is a root → a−bia - bia−bi is also a root
• If 2\sqrt{2}2​ is a root → −2-\sqrt{2}−2​ is also a root

Number of Roots

A polynomial of degree n has:

• Exactly n total roots
• Real or complex

Pascal’s Triangle

Used for Binomial Expansion.

Core Construction

• Starting Point: The triangle begins with a single 1

• The Rule: Each subsequent row starts and ends with 1. Every interior number is the sum of the two numbers immediately above it.

Row numbers start at 0.

Example rows:

Row 0: 1
Row 1: 1 1
Row 2: 1 2 1
Row 3: 1 3 3 1
Row 4: 1 4 6 4 1

Row 5: 1 5 10 10 5 1

Row 6: 1 6 15 20 15 6 1

Row 7: 1 7 21 35 35 21 7 1

Row 8: 1 8 28 56 70 56 70 56 28 8 1

Binomial Theorem

(a+b)n(a + b)^n(a+b)n

Use Pascal’s Triangle coefficients.

Example:

(x+1)3(x + 1)^3(x+1)3

Coefficients: 1 3 3 1

x3+3x2+3x+1x^3 + 3x^2 + 3x + 1x3+3x2+3x+1

Finding a Specific Term

General term:

(nk)an−kbk\binom{n}{k} a^{n-k} b^k(kn​)an−kbk

Polynomial Unit Practice Test

Part 1: Standard Form & End Behavior

1. Write in standard form. State the degree and end behavior.
y=5−2x3+x4−7xy = 5 - 2x^3 + x^4 - 7xy=5−2x3+x4−7x

2. Write in standard form. Classify by degree.
y=3x2−9+4x5y = 3x^2 - 9 + 4x^5y=3x2−9+4x5

Part 2: Factoring & Solving

3. Solve by factoring:
x2−9x+20=0x^2 - 9x + 20 = 0x2−9x+20=0

4. Solve:
x3−4x=0x^3 - 4x = 0x3−4x=0

5. Solve:
2x2−8=02x^2 - 8 = 02x2−8=0

Part 3: Zeros & Multiplicity

6. Find the zeros and multiplicity:
y=(x−1)2(x+3)3y = (x - 1)^2(x + 3)^3y=(x−1)2(x+3)3

Will the graph cross or bounce at each zero?

Part 4: Synthetic Division

7. Divide using synthetic division:

(x3−2x2−5x+6)÷(x−3)(x^3 - 2x^2 - 5x + 6) \div (x - 3)(x3−2x2−5x+6)÷(x−3)

Part 5: Remainder Theorem

8. Find the remainder when

P(x)=x3+2x2−7x+4P(x) = x^3 + 2x^2 - 7x + 4P(x)=x3+2x2−7x+4

is divided by x−2x - 2x−2

Part 6: Rational Root Theorem

9. List all possible rational zeros of:

P(x)=2x3+3x2−8x+4P(x) = 2x^3 + 3x^2 - 8x + 4P(x)=2x3+3x2−8x+4

Part 7: Descartes’ Rule of Signs

10. Determine the possible number of positive and negative real zeros of:

P(x)=x4−3x3+x2+5P(x) = x^4 - 3x^3 + x^2 + 5P(x)=x4−3x3+x2+5

Part 8: Binomial Expansion

11. Expand:

(x+2)3(x + 2)^3(x+2)3

12. Find the coefficient of the x2x^2x2 term in:

(3x−1)4(3x - 1)^4(3x−1)4