91Ó°ÊÓ

Polynomials are mathematical expressions that consist of variables, coefficients, and exponents. The general form of a polynomial is expressed as:

• \( a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \)

Here, \(a_n, a_{n-1}, \cdots, a_1, a_0\) are constants, and \(n\) is a non-negative integer that represents the highest degree of the polynomial.
The degree of a polynomial indicates the highest power of the variable present in the expression.
A quadratic function, for example, is a polynomial of degree 2, typically written in the form \(ax^2 + bx + c\). Polynomials can be used to model a wide range of real-world scenarios and are fundamental to algebra and calculus.
Knowing how to express functions in polynomial form is crucial for solving equations analytically and understanding the behavior of graphs.

Zeros, also known as roots, of a function are the values of \(x\) for which the function outputs zero.

• For a quadratic function in the form \(f(x) = ax^2 + bx + c\), the zeros occur where \(f(x) = 0\).
• These points are important because they indicate where the graph of the function intersects the x-axis.
• In this exercise's context, the zeros given are -4 and 9, which are the solutions to the equation \(a(x + 4)(x - 9) = 0\).

To find the quadratic function with these zeros, we used the formula \(f(x) = a(x + 4)(x - 9)\).
Understanding zeros is essential in solving quadratic equations, graphing quadratic functions, and analyzing the property of the graphs.

Quadratic equations are a specific type of polynomial equation where the highest degree of the variable is 2. The standard form is:

• \(ax^2 + bx + c = 0\)

These equations can have up to two real solutions, also known as the equation's roots or zeros.
To solve a quadratic equation, you can apply several methods, such as factoring, using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), and completing the square. In this exercise, we factored the quadratic functions by recognizing the zeros: -4 and 9.
Furthermore, we demonstrate using known values, like a point on the graph, to determine additional constants, such as the coefficient \(a\). With the point (3, 5) provided in the exercise, it helped us solve for \(a\) and consequently determine the exact form of the quadratic equation: \(-\frac{5}{42}x^2 + \frac{25}{42}x + \frac{30}{7}\).Quadratic equations are fundamental in calculus and physics, modeling various phenomena such as projectile motion and optimization problems.