Study sheet: Polynomial Functions and Theorems

📋 Course Outline

1. Polynomial Function Structure
2. Recognizing Polynomial Functions
3. Rational Root Theorem
4. Remainder Theorem
5. Factor Theorem and Zeros

📖 1. Polynomial Function Structure

🔑 Key Concepts & Definitions

• Polynomial expression : A polynomial expression in x of degree n has the form $a_nx^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_1x+a_0$, where $a_n\ne0$, n is a nonnegative integer, and all coefficients are real numbers.

📝 Essential Points

📌 For a polynomial expression, the degree n is a nonnegative integer and the leading coefficient a_n is not zero.

📖 2. Recognizing Polynomial Functions

★ Must-know

• A polynomial function cannot contain:
  • a negative exponent
  • a fractional exponent
  • a variable in the denominator
  • a variable inside a radical
  • a decimal exponent

Further detail

• The expression $2x^{-2}+3x+5$ is not a polynomial because it contains a negative exponent.

• The expression $2x^{1/2}+3x+5$ is not a polynomial because it contains a fractional exponent.

• An expression with a variable in the denominator or inside a radical is not a polynomial function.

💡 Memory Hook

Polynomial terms use whole-number exponents; non-polynomial terms use forbidden exponents or variable denominators.

📖 3. Rational Root Theorem

🔑 Key Concepts & Definitions

• Rational Root Theorem : The Rational Root Theorem states that for a polynomial equation with integer coefficients, any rational root written in lowest terms as $\frac{p}{q}$ must have p as a factor of the constant term and q as a factor of the leading coefficient.

★ Must-know

📐 Formula — For $f(x)=a_nx^n+\cdots+a_1x+a_0$, the possible rational zeros have the form $x=\frac{p}{q}$, where p is a factor of the constant term and q is a factor of the leading coefficient.

Further detail

• To apply the Rational Root Theorem, identify the constant term, identify the leading coefficient, list their factors, and form all possible values of $p/q$.

💡 Memory Hook

Constant factors p, leading-coefficient factors q, possible roots p/q.

📖 4. Remainder Theorem

🔑 Key Concepts & Definitions

• Remainder Theorem : The Polynomial Remainder Theorem states that when a polynomial f(x) is divided by the linear factor x−c, the remainder equals f(c).

★ Must-know

• 🔄 To use the theorem:
  1. choose the candidate c
  2. substitute c into f(x)
  3. evaluate f(c)

Further detail

• For $f(x)=x^3-6x^2+11x-6$, evaluating at x=1 gives $f(1)=0$.

💡 Memory Hook

Evaluate f(c) → obtain the remainder when dividing by x − c.

📖 5. Factor Theorem and Zeros

🔑 Key Concepts & Definitions

• Factor Theorem : The Factor Theorem states that x−c is a factor of f(x) if and only if f(c)=0.

★ Must-know

📌 A zero c corresponds to the factor x−c, so a positive zero 1 gives the factor x−1 while a negative zero −1 gives the factor x+1.

• For $f(x)=x^3-6x^2+11x-6$, the tested values 1, 2, and 3 give a remainder of zero, so the zeros are x=1, x=2, and x=3.

Further detail

• For $f(x)=x^3-6x^2+11x-6$, f(−1)=−24, f(−2)=−60, f(−3)=−120, f(6)=60, and f(−6)=−504, so x+1, x+2, x+3, x−6, and x+6 are not factors.

💡 Memory Hook

If f(c)=0 → x−c is a factor → c is a zero.

📊 Synthesis Tables

Root and Factor Tests

Theorem	What it determines	Condition
Rational Root Theorem	Possible rational zeros	p is a factor of the constant term and q is a factor of the leading coefficient
Remainder Theorem	Remainder after division by x−c	Remainder = f(c)
Factor Theorem	Whether x−c is a factor	f(c)=0