Unit 4 Precalc

4.1: Polynomial Functions + Their Properties

Key Terms

• Polynomial function: f(x)=anxn+...+a1x+a0f(x) = a_nx^n + ... + a_1x + a_0f(x)=an​xn+...+a1​x+a0​

• Degree: highest power of xxx

• Leading coefficient: the coefficient of the highest-power term

General Shape Rules

• Even degree, positive LC: up ↑, down ↑

• Even degree, negative LC: down ↓, down ↓

• Odd degree, positive LC: down ↓, up ↑

• Odd degree, negative LC: up ↑, down ↓

End Behavior

Use leading term: f(x)≈anxnf(x) \approx a_n x^nf(x)≈an​xn

Turning Points

• Max number = n−1n - 1n−1

Key Skills

• Classify functions by degree

• Determine end behavior

• Identify local maxima/minima

• Recognize symmetry (even, odd)

4.2: Graphing Polynomial Functions

Graphing Strategy

1. Find zeros (x-intercepts)

2. Determine multiplicity

   • Odd → crosses x-axis

   • Even → touches/bounces

3. Check end behavior

4. Plot y-intercept

5. Sketch smooth curve

Multiplicity Behavior

• 1 → linear cross

• 2 → bounce

• 3 → flatten-cross

• Larger → increasingly “flat”

Key Concepts

• Intermediate Value Theorem: confirms existence of a zero in interval

• Smooth + continuous graphs (no breaks)

4.3: Properties of Rational Functions

Form: P(x)/Q(x)

Vertical asymptotes

• Where Q(x)=0Q(x)=0Q(x)=0 and doesn’t cancel.

Holes

• Factor cancels → hole at that x-value.

Horizontal / Slant Asymptotes

• Compare degrees of numerator (N) and denominator (D):

Behavior near vertical asymptotes

• Look at left/right limits.

4.5: Polynomial & Rational Inequalities

Polynomial Inequalities

1. Move everything to one side

2. Factor

3. Set up a sign chart

4. Test intervals

5. Write solution in interval notation

Rational Inequalities

1. Bring to one side

2. Factor numerator & denominator

3. Critical points = zeros of numerator + denominator

4. Sign chart (exclude points where denominator = 0)

5. Choose intervals

Golden Rule

Never multiply both sides by a variable expression — signs can flip.
Use sign charts instead.

4.6: Real Zeros of Polynomial Functions

Rational Zero Test

Possible rational zeros:

±factors of constantfactors of leading coeff\pm \frac{\text{factors of constant}}{\text{factors of leading coeff}}±factors of leading coefffactors of constant​

Synthetic Division

Used to:

• Test zeros

• Factor polynomials

• Find depressed polynomial

Fundamental Theorem of Algebra

Degree nnn → exactly nnn complex zeros including multiplicity.

Descartes’ Rule of Signs

Positive real zeros → # of sign changes
Negative real zeros → sign changes of f(−x)

4.7: Complex Zeros

Imaginary & Complex Roots

If a polynomial has real coefficients, complex roots come in conjugate pairs:

• If a+bia + bia+bi is a root → a−bia - bia−bi is also a root

Quadratic roots with discriminant

• b2−4ac>0b^2 - 4ac > 0b2−4ac>0: 2 real

• b2−4ac=0b^2 - 4ac = 0b2−4ac=0: 1 real (double root)

• b2−4ac<0b^2 - 4ac < 0b2−4ac<0: 2 complex

Finding Complex Roots

• Synthetic division with a known real root → factor to quadratic → quadratic formula

• Or directly solve via factoring

1. Vertical Asymptotes

Definition: Vertical asymptotes are vertical lines x=a where the function “blows up” (goes to ±∞).

How to find them:

1. Set the denominator equal to 0:

2. Solve for x.

3. Check for holes first: If the same factor appears in both the numerator and denominator, that factor gives a hole, not an asymptote. Only factors left in the denominator after simplification give vertical asymptotes.

2. X-Intercepts (Zeros)

Definition: Points where the graph crosses the x-axis 

How to find them:

1. Set the numerator equal to 0:

2. Solve for x

3. Make sure the x-value does not make the denominator 0 (if it does, it’s a hole, not an x-intercept).

3. Horizontal Asymptotes

How to find them: Compare the degrees of the numerator (nnn) and denominator (mmm):

Purpose of Using Synthetic Division Here

1. Check if a given number is a zero of a polynomial

   • We were told x=2 is a zero.

   • Synthetic division confirms that when we divide the polynomial by (x-2) the remainder is 0. ✅

2. Reduce the degree of the polynomial to quadratic 

3. Find all other zeros of the polynomial