Quadratic Equations and Polynomial Functions: Study Guide

Quadratic Equations

Solving Quadratic Equations

Quadratic equations are equations of the form ax2 + bx + c = 0. There are several methods to solve them, including factoring, completing the square, and using the quadratic formula.

• Factoring: Express the quadratic as a product of two binomials and set each factor to zero.

• Quadratic Formula: For any quadratic equation ax2 + bx + c = 0, the solutions are given by: $x = \frac{ -b \pm \sqrt{ b^2 - 4ac } }{ 2a }$

• Completing the Square: Rewrite the equation in the form (x - h)2 = k and solve for x.

• Example: Solve $x^2 - 3x + 2 = 0$ by factoring: $(x-1)(x-2) = 0$ so $x = 1$ or $x = 2$.

Vertex of a Quadratic Function

The vertex of a quadratic function f(x) = ax2 + bx + c is the highest or lowest point on its graph (a parabola). The vertex can be found using the formula:

• Vertex Formula: $x_v = -\frac{b}{2a}$

• Substitute $x_v$ into the function to find $y_v$.

• Example: For $h(x) = x^2 - 10x + 25$, $a = 1$, $b = -10$, so $x_v = 5$, $y_v = h(5) = 0$. Vertex: (5, 0).

Graphing Quadratic Functions

To graph a quadratic function, identify the vertex, axis of symmetry, y-intercept, x-intercepts, and direction of opening (upward if a > 0, downward if a < 0).

• Vertex: Use the vertex formula above.

• Y-intercept: Set $x = 0$ and solve for $f(0)$.

• X-intercepts: Set $f(x) = 0$ and solve for $x$.

• Direction: If $a > 0$, the parabola opens upward; if $a < 0$, it opens downward.

• Example: $g(x) = 2(x+2)^2 - 2$ has vertex at $(-2, -2)$, opens upward, y-intercept at $g(0) = 2(0+2)^2 - 2 = 6$.

Polynomial Functions and Synthetic Division

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form (x - k). It is especially useful for finding zeros and evaluating polynomials.

• Steps:

  1. Write the coefficients of the polynomial.

  2. Write the zero k of the divisor (for x - k).

  3. Bring down the leading coefficient.

  4. Multiply by k, add to the next coefficient, and repeat.

• Example: Divide $2x^5 + x^4 - 10x^3 + 2x^2 + x - 16$ by $(x + 2)$ using k = -2.

Testing for Zeros Using Synthetic Division

To determine if a number k is a zero of a polynomial f(x), use synthetic division. If the remainder is zero, then k is a zero of f(x); otherwise, the remainder is f(k).

• Example: If dividing by $(x - 1)$ yields a remainder of 0, then 1 is a zero of the polynomial.

Rational Zero Theorem

The Rational Zero Theorem helps to list all possible rational zeros of a polynomial function. Possible rational zeros are of the form $\pm \frac{p}{q}$, where p divides the constant term and q divides the leading coefficient.

• Example: For $g(x) = x^3 - 3x^2 - 13x + 15$, possible rational zeros are $\pm1, \pm3, \pm5, \pm15$.

Graphing Polynomial Functions

To graph a polynomial function, identify its degree, leading coefficient, zeros, and end behavior. Plot the zeros and analyze the behavior near each zero (crosses or touches the x-axis).

• Example: $f(x) = -x^3 - 5x^2 + x + 5$ is a cubic function opening downward (since leading coefficient is negative).

Summary Table: Key Features of Quadratic and Polynomial Functions

Feature	Quadratic Function	Polynomial Function
General Form	$ax^2 + bx + c$	$a_nx^n + \ldots + a_1x + a_0$
Degree	2	n (n ≥ 1)
Vertex	Yes	No (unless n = 2)
Axis of Symmetry	Yes	No (unless n = 2)
Number of Zeros	Up to 2	Up to n
End Behavior	Up or down	Depends on degree and leading coefficient