91Ó°ÊÓ

X-intercepts of a polynomial are the points where the graph of the function crosses the x-axis. These occur where the value of the polynomial is zero, meaning that the height of the function is zero at these points.
When you have a polynomial given in factored form, such as \( f(t) = 2(t-1)(t+2)(t-3) \), finding the x-intercepts becomes straightforward because you can directly see the factors involved.
To find the x-intercepts, set the whole equation equal to zero:

• \( 2(t-1)(t+2)(t-3) = 0 \)

This leads us to solve for \( t \) by setting each factor separately equal to zero:

• \( t-1=0 \implies t=1 \)
• \( t+2=0 \implies t=-2 \)
• \( t-3=0 \implies t=3 \)

Thus, the x-intercepts are \( t = 1, -2, \) and \( 3 \). These are the values where the function touches or crosses the x-axis. At these points, the output, or \( f(t) \), indeed equals zero.

Y-intercepts are the points where the graph of the function crosses the y-axis. These occur when the input value, represented typically by \( x \) or \( t \) in function notation, is equal to zero. In simpler terms, the y-intercept reflects where the function lands vertically when you are at the origin horizontally.
To find the y-intercept, simply plug in zero for \( t \) in the function provided. If the function is \( f(t) = 2(t-1)(t+2)(t-3) \), you calculate the y-intercept as follows:

• Substitute \( t = 0 \) into the function: \( f(0) = 2(0-1)(0+2)(0-3) \)
• Calculate: \( f(0) = 2(-1)(2)(-3) = 2(-6) = -12 \)

The y-intercept of this function is \( (0, -12) \), meaning that the graph crosses the y-axis 12 units below the origin.

Factored form is a way of expressing a polynomial as a product of its factors, which are simpler polynomials. This form is especially useful because it easily reveals the roots or solutions of the polynomial. A function is said to be in factored form if it looks like \( a(x-x_1)(x-x_2)...(x-x_n) \), where \( a \) is a constant and \( x_1, x_2, ..., x_n \) are the roots of the polynomial.
In our exercise, the function \( f(t) = 2(t-1)(t+2)(t-3) \) is already given in factored form. This means

• The multiplication expands out to a polynomial, but keeping it factored helps quickly find the x-intercepts or roots.
• Each bracketed term represents a point on the graph where \( f(t) \) equals zero, also called a "zero" or "root."

Advantages of using the factored form include:

• It immediately shows the x-intercepts of the polynomial.
• It helps in sketching the graph by giving you a visual on where the polynomial crosses the x-axis.
• Simplifies the process of solving polynomial equations by direct observation.

Overall, the factored form is a powerful format that reveals much about the polynomial's behavior and makes certain calculations much faster.