91Ó°ÊÓ

Factoring polynomials is a key step in finding the zeros of a function.
A polynomial can often be broken down into simpler 'factors' that, when multiplied together, give the original polynomial.
For example, in the function given, \(r(x) = x^3 + 4x^2 + 3x\), we start by factoring out the common term, which is \(x\). This gives us \(r(x) = x(x^2 + 4x + 3)\).
The next step is to factor the quadratic expression within the parentheses further. To do this, we look for two numbers that multiply to the constant term (3) and add up to the linear coefficient (4).
Those numbers are 3 and 1, so we can write \(x^2 + 4x + 3\) as \( (x + 1)(x + 3)\). This gives us the fully factored form \(r(x) = x(x + 1)(x + 3)\).

A quadratic equation is a polynomial of degree 2, and has the form \(ax^2 + bx + c\).
Solving quadratic equations is crucial for factoring polynomials since quadratic terms often appear in more complex polynomials.
Common methods to solve quadratic equations include factoring (as we did above), using the quadratic formula, and completing the square.
In the example given, \(x^2 + 4x + 3\) is a quadratic equation. By factoring, we found the roots \(x + 1\) and \(x + 3\), effectively solving the equation.
Remember, solving a quadratic equation can give us the zeros of the quadratic part of the polynomial.

The roots or zeros of functions are the values of \(x\) for which the function equals zero.
Finding these zeros helps us understand the behavior of the function, such as where it crosses the x-axis.
To find the zeros of a polynomial, we set the fully factored form equal to zero and solve for \(x\).
In our example, we set each factor of \(r(x) = x(x + 1)(x + 3)\) equal to zero: \(x = 0, x + 1 = 0, x + 3 = 0\).
Solving these, we find that \(x = 0\), \(x = -1\), and \(x = -3\).
These are the zeros of the polynomial function \(r(x)\). Knowing the roots allows us to graph the polynomial and understand its intersections with the x-axis.