Teach me everything on polynomials and quadratic equations

Let's break down polynomials and quadratic equations.

Polynomials

A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

• General Form: A polynomial in one variable $x$ can be written as:

  $$P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$$

  where $a_n, a_{n-1}, \dots, a_0$ are coefficients (real numbers), $x$ is the variable, and $n$ is a non-negative integer called the degree of the polynomial (if $a_n \neq 0$).

• Terms:

  • Monomial: A polynomial with one term (e.g., $5x^3$).
  • Binomial: A polynomial with two terms (e.g., $2x^2 - 7$).
  • Trinomial: A polynomial with three terms (e.g., $x^2 + 3x - 1$).

• Degree: The highest exponent of the variable in a polynomial.

  • Example: $3x^4 - 2x^2 + x - 5$ has a degree of 4.
  • Linear polynomial: Degree 1 (e.g., $2x + 3$).
  • Quadratic polynomial: Degree 2 (e.g., $x^2 - 4x + 1$).
  • Cubic polynomial: Degree 3 (e.g., $x^3 + 2x^2 - x + 5$).

• Standard Form: Writing terms in descending order of their degrees.

  • Example: $5x - 2x^3 + 7$ in standard form is $-2x^3 + 5x + 7$.

• Operations: Polynomials can be added, subtracted, and multiplied.

  • Example (Multiplication): $(x+2)(x-3) = x^2 - 3x + 2x - 6 = x^2 - x - 6$.

• Roots/Zeros: The values of $x$ for which $P(x) = 0$. These are the x-intercepts of the polynomial's graph.

Quadratic Equations

A quadratic equation is a polynomial equation of degree 2.

• Standard Form:

  $$ax^2 + bx + c = 0$$

  where $a, b, c$ are real numbers and $a \neq 0$.

• Methods to Solve Quadratic Equations:

  1. Factoring: If the quadratic expression can be factored, set each factor to zero and solve for $x$.

     • Example: Solve $x^2 - 5x + 6 = 0$. $$(x-2)(x-3) = 0$$ $$x-2=0 \implies x=2$$ $$x-3=0 \implies x=3$$ The solutions are $x=2$ and $x=3$.

  2. Completing the Square: A method to convert a quadratic expression into a perfect square trinomial. This can be used to derive the quadratic formula.

  3. Quadratic Formula: This formula can solve any quadratic equation. For $ax^2 + bx + c = 0$, the solutions for $x$ are given by:

     $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
     • Example: Solve $2x^2 + 3x - 2 = 0$. Here, $a=2$, $b=3$, $c=-2$. $$x = \frac{-(3) \pm \sqrt{(3)^2 - 4(2)(-2)}}{2(2)}$$ $$x = \frac{-3 \pm \sqrt{9 + 16}}{4}$$ $$x = \frac{-3 \pm \sqrt{25}}{4}$$ $$x = \frac{-3 \pm 5}{4}$$ Two solutions: $$x_1 = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{1}{2}$$ $$x_2 = \frac{-3 - 5}{4} = \frac{-8}{4} = -2$$ The solutions are $\boxed{x = \frac{1}{2}, -2}$.

• The Discriminant ($\Delta$): The expression under the square root in the quadratic formula, $\Delta = b^2 - 4ac$. It tells us about the nature of the roots:

  • If $\Delta > 0$: Two distinct real roots.
  • If $\Delta = 0$: One real root (a repeated root).
  • If $\Delta < 0$: Two complex conjugate roots (no real roots).

• Graphing Quadratic Equations (Parabolas): The graph of a quadratic equation $y = ax^2 + bx + c$ is a parabola.

  • If $a > 0$, the parabola opens upwards.
  • If $a < 0$, the parabola opens downwards.
  • The vertex is the turning point of the parabola. Its x-coordinate is given by $x = -\frac{b}{2a}$.
  • The axis of symmetry is the vertical line passing through the vertex, $x = -\frac{b}{2a}$.