Page 2: Transforming Polynomials and Identities

This page delves into the process of transforming polynomials into canonical form and explores important algebraic identities.

To write a trinomial in canonical form:

1. Factor out the coefficient of x²
2. Complete the square
3. Use the identity $a - b$² = a² - 2ab + b²

Example: Let's transform g(x) = 2x² - 20x + 10 into canonical form:

1. g(x) = 2$x² - 10x$ + 10
2. g(x) = 2$(x - 5)² - 25$ + 10
3. g(x) = 2$x - 5$² - 50 + 10
4. g(x) = 2$x - 5$² - 40

This page also highlights important algebraic identities:

Vocabulary:

• $a + b$² = a² + 2ab + b²
• $a - b$² = a² - 2ab + b²

These identities are crucial for manipulating and simplifying algebraic expressions, especially when dealing with second degree polynomials.

The page concludes by noting that to find the canonical form a$x - α$² + β from ax² + bx + c, one can simply develop the expression and equate coefficients.

Page 3: Sign Analysis of Second Degree Polynomials

This final page focuses on analyzing the sign of second degree polynomials, which is crucial for solving inequalities and studying function behavior.

The sign analysis depends on the discriminant (Δ):

1. If Δ < 0: The polynomial has the same sign as 'a' for all real x.

2. If Δ = 0: The polynomial has the same sign as 'a' everywhere except at x₀ = -b/(2a), where it equals zero.

3. If Δ > 0: The polynomial changes sign at its roots. Between the roots, it has the opposite sign of 'a'. Outside the roots, it has the same sign as 'a'.

Highlight: The sign of 'a' (the coefficient of x²) determines whether the parabola opens upward (a > 0) or downward (a < 0).

The page includes a visual representation of these cases, showing how the graph of the polynomial behaves in each scenario.

Example: For a polynomial with Δ > 0 and a > 0, the function is negative between its roots and positive elsewhere.

This sign analysis is particularly useful when solving second degree inequalities or studying the behavior of quadratic functions in different intervals.

Page 1: Solving Second Degree Equations and Polynomial Forms

This page focuses on solving second degree equations and understanding different forms of polynomials.

The discriminant (Δ) plays a crucial role in solving second degree equations. The number and nature of solutions depend on its value. For an equation ax² + bx + c = 0:

Definition: The discriminant is given by Δ = b² - 4ac.

• If Δ < 0, the equation has no real solutions.
• If Δ = 0, there is one unique solution: x = -b / (2a).
• If Δ > 0, there are two distinct solutions: x = $-b ± √Δ$ / (2a).

The page also introduces two important forms of second degree polynomials:

1. Factored form: f(x) = a$x - x₁$$x - x₂$
2. Canonical form: f(x) = a$x - α$² + β

Highlight: The canonical form is particularly useful for studying the function's behavior.

For a > 0, the function has a minimum at x = α, with the minimum value being β. Conversely, for a < 0, it has a maximum at x = α, with the maximum value being β.

Example: In the canonical form f(x) = a$x - α$² + β, α represents the x-coordinate of the vertex, and β represents the y-coordinate.