Pre-Calculus - Week 2

Polynomial Functions

Polynomials are the workhorses of algebra and calculus. This week you master quadratic functions in vertex form, analyze the end behavior and zeros of higher-degree polynomials, apply the Factor Theorem, and sketch polynomial graphs by hand. These skills recur constantly in Calculus I when you differentiate and integrate polynomials.

Learning Objectives

- Write quadratic functions in vertex form and identify the vertex. - Solve quadratic equations by factoring, completing the square, and the quadratic formula. - Find real zeros of polynomials using the Factor Theorem. - Determine end behavior of a polynomial from its leading term. - Sketch a polynomial graph using zeros, multiplicity, and end behavior. - Apply the Intermediate Value Theorem to locate real zeros. - Perform polynomial long division and synthetic division.

1. Quadratic Functions and Vertex Form

A quadratic function has the form f(x) = ax^2 + bx + c where a is not 0. Vertex form: f(x) = a(x - h)^2 + k. Vertex is (h, k). If a > 0 the parabola opens up (minimum at vertex). If a < 0 the parabola opens down (maximum at vertex). Example 1: f(x) = 2(x - 3)^2 + 1. Vertex (3, 1), opens up. Example 2: Convert f(x) = x^2 - 6x + 5 to vertex form. Complete the square: x^2 - 6x + 9 - 9 + 5 = (x - 3)^2 - 4. Vertex (3, -4), opens up, minimum value -4.

2. Solving Quadratics: Factoring

If ax^2 + bx + c = 0 can be factored, set each factor equal to zero. Example 1: Solve x^2 - 5x + 6 = 0. Factor: (x - 2)(x - 3) = 0. Solutions: x = 2 or x = 3. Example 2: Solve 2x^2 + 5x - 3 = 0. Factor: (2x - 1)(x + 3) = 0. Solutions: x = 1/2 or x = -3. Example 3: Solve 3x^2 = 12x. Rearrange: 3x^2 - 12x = 0. Factor: 3x(x - 4) = 0. Solutions: x = 0 or x = 4.

3. The Quadratic Formula

For ax^2 + bx + c = 0: x = (-b ± sqrt(b^2 - 4ac)) / (2a) The discriminant b^2 - 4ac tells you how many real solutions: - b^2 - 4ac > 0: two distinct real solutions - b^2 - 4ac = 0: one repeated real solution - b^2 - 4ac < 0: no real solutions (two complex solutions) Example: Solve x^2 + 4x + 1 = 0. a=1, b=4, c=1. x = (-4 ± sqrt(16-4))/2 = (-4 ± sqrt(12))/2 = -2 ± sqrt(3). Solutions: x = -2 + sqrt(3) and x = -2 - sqrt(3).

4. End Behavior of Polynomials

End behavior is what f(x) does as x goes to +infinity or -infinity. Only the leading term determines end behavior. Four cases: 1. Even degree, positive leading coefficient: both ends go UP. 2. Even degree, negative leading coefficient: both ends go DOWN. 3. Odd degree, positive leading coefficient: left DOWN, right UP. 4. Odd degree, negative leading coefficient: left UP, right DOWN. Example 1: f(x) = 3x^4 - 2x + 1. Even, positive. Both ends up. Example 2: f(x) = -2x^3 + x. Odd, negative. Left up, right down.

5. Zeros and Multiplicity

A zero of f is an x where f(x) = 0. Each zero has a multiplicity. - Odd multiplicity: the graph CROSSES the x-axis at that zero. - Even multiplicity: the graph TOUCHES the x-axis and turns back (bounces). Example: f(x) = (x - 1)^2 (x + 2)(x - 3)^3. Zero at x = 1: multiplicity 2 (even) — bounces. Zero at x = -2: multiplicity 1 (odd) — crosses. Zero at x = 3: multiplicity 3 (odd) — crosses (with a flatter S-curve).

6. The Factor Theorem

The Factor Theorem: (x - c) is a factor of f(x) if and only if f(c) = 0. This means: if you suspect x = c is a zero, evaluate f(c). If f(c) = 0, then (x - c) is a factor. Example: Show that x = 2 is a zero of f(x) = x^3 - 3x^2 + 4. f(2) = 8 - 12 + 4 = 0. YES, (x - 2) is a factor. Now divide f(x) by (x - 2) to find the other factors: f(x) = (x - 2)(x^2 - x - 2) = (x - 2)(x - 2)(x + 1) = (x - 2)^2(x + 1). All zeros: x = 2 (mult. 2) and x = -1 (mult. 1).

7. Polynomial Long Division

Divide polynomials the same way as long division with integers. Example: Divide x^3 + 2x^2 - x - 2 by (x - 1). Step 1: x^3 / x = x^2. Multiply: x^2(x - 1) = x^3 - x^2. Subtract: 3x^2 - x - 2. Step 2: 3x^2 / x = 3x. Multiply: 3x(x-1) = 3x^2 - 3x. Subtract: 2x - 2. Step 3: 2x / x = 2. Multiply: 2(x-1) = 2x - 2. Subtract: 0. Result: x^2 + 3x + 2 = (x + 1)(x + 2). So f(x) = (x - 1)(x + 1)(x + 2). Zeros: x = 1, -1, -2.

8. Synthetic Division

Synthetic division is a shortcut for dividing by a linear factor (x - c). Example: Divide f(x) = 2x^3 - 3x^2 - 8x + 12 by (x - 2). Write coefficients: 2, -3, -8, 12. Use c = 2. 2 | 2 -3 -8 12 | 4 2 -12 | 2 1 -6 0 Quotient: 2x^2 + x - 6. Remainder: 0. Factor: 2x^2 + x - 6 = (2x - 3)(x + 2). So f(x) = (x - 2)(2x - 3)(x + 2). Zeros: x = 2, 3/2, -2.

9. Sketching Polynomial Graphs

To sketch a polynomial f(x): 1. Find and plot all real zeros (and their multiplicities). 2. Determine end behavior from the leading term. 3. Check whether the graph crosses (odd mult.) or bounces (even mult.) at each zero. 4. Plot a few additional points if needed. 5. Connect smoothly — polynomial graphs have no sharp corners. Example: Sketch f(x) = x(x - 2)^2. Zeros: x = 0 (mult. 1, crosses), x = 2 (mult. 2, bounces). Leading term: x^3. Odd degree, positive: left down, right up. At x = 1: f(1) = 1(1)^2 = 1. Plot (1, 1).

10. Intermediate Value Theorem

The Intermediate Value Theorem (IVT): If f is continuous on [a, b] and N is any number between f(a) and f(b), then there exists at least one c in (a, b) where f(c) = N. For zeros: if f(a) and f(b) have opposite signs, there is at least one zero between a and b. Example: f(x) = x^3 - 4x + 1. f(0) = 1 > 0, f(1) = -2 < 0. By IVT, there is a zero in (0, 1). f(-2) = -8 + 8 + 1 = 1 > 0, f(-3) = -27 + 12 + 1 = -14 < 0. Zero in (-3, -2).

11. Common Mistakes

- Forgetting that a factor (x - c)^2 means a BOUNCE, not a cross. - Confusing end behavior: odd degree with positive leading coefficient goes left-down, right-up (like the basic x^3 shape). - Stopping after factoring — always find ALL zeros including multiplicity. - Using the quadratic formula but making a sign error in -b or in the discriminant. - Forgetting that the leading COEFFICIENT (not just degree) determines end behavior direction for odd degree.