X-Intercepts Of Polynomial Function: A Step-by-Step Guide

Nov 11, 2025 by Admin 58 views

Let's dive into finding the x-intercepts of the polynomial function f(x) = x^4 - 2x^3 - 18x^2 + 6x + 45, especially focusing on where the function crosses the x-axis. We'll express these intercepts as ordered pairs. This might seem daunting, but don't worry, we'll break it down into manageable steps. So, grab your favorite beverage, and let's get started!

Understanding X-Intercepts

X-intercepts, also known as roots or zeros, are the points where the graph of a function intersects the x-axis. At these points, the value of the function, f(x), is equal to zero. Finding these intercepts is crucial in understanding the behavior of the polynomial function. Essentially, we're solving the equation f(x) = 0.

For our specific polynomial f(x) = x^4 - 2x^3 - 18x^2 + 6x + 45, we need to find the values of x that make this equation true. Now, solving a quartic equation (an equation with the highest power of x being 4) can be tricky, but we can use several techniques such as factoring, synthetic division, or using a graphing calculator to help us out. Factoring, if possible, is often the cleanest approach, but it might not always be straightforward.

One common strategy is to look for rational roots using the Rational Root Theorem. This theorem states that any rational root of the polynomial must be a factor of the constant term (45 in our case) divided by a factor of the leading coefficient (1 in our case). So, possible rational roots are ±1, ±3, ±5, ±9, ±15, and ±45. We can test these values by plugging them into the function to see if any of them make f(x) = 0. Alternatively, we can use synthetic division to test these potential roots more efficiently. When a potential root r is tested and results in a remainder of zero, then r is indeed a root, and (x - r) is a factor of the polynomial.

Another approach is to use a graphing calculator or software to visualize the polynomial. By plotting the graph, we can visually identify the x-intercepts, which can give us a good starting point for further analysis. Keep in mind that a graph might not always give exact values, especially if the roots are irrational, but it can help narrow down the possibilities and guide our algebraic methods.

Factoring the Polynomial

Factoring is the key to finding the x-intercepts in a straightforward manner. The given polynomial is f(x) = x^4 - 2x^3 - 18x^2 + 6x + 45. Let's attempt to factor it. Factoring this polynomial directly might be challenging. However, upon closer inspection or by using a bit of trial and error (or a graphing calculator), we can find that x = -3 and x = 3 are roots of the polynomial. This means (x + 3) and (x - 3) are factors.

So, let's perform polynomial division to divide f(x) by (x + 3) and (x - 3). First, we can multiply these factors to get (x + 3)(x - 3) = x^2 - 9. Now, we divide f(x) by x^2 - 9. After performing the polynomial long division, we find that:

f(x) = (x^2 - 9)(x^2 - 2x - 5)

Now we have factored the polynomial into a product of two quadratic expressions. The first quadratic, x^2 - 9, factors further into (x - 3)(x + 3), as we already knew. The second quadratic, x^2 - 2x - 5, doesn't factor easily using integers, so we'll need to use the quadratic formula to find its roots.

The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

For the quadratic x^2 - 2x - 5, we have a = 1, b = -2, and c = -5. Plugging these values into the quadratic formula, we get:

x = (2 ± √((-2)^2 - 4(1)(-5))) / (2(1)) x = (2 ± √(4 + 20)) / 2 x = (2 ± √24) / 2 x = (2 ± 2√6) / 2 x = 1 ± √6

So, the roots of the quadratic x^2 - 2x - 5 are 1 + √6 and 1 - √6. Therefore, the complete factorization of f(x) is:

f(x) = (x - 3)(x + 3)(x - (1 + √6))(x - (1 - √6))

Finding the X-Intercepts

X-intercepts occur where f(x) = 0. From our factorization, we have:

(x - 3)(x + 3)(x - (1 + √6))(x - (1 - √6)) = 0

This gives us the following x-intercepts:

x - 3 = 0 => x = 3
x + 3 = 0 => x = -3
x - (1 + √6) = 0 => x = 1 + √6
x - (1 - √6) = 0 => x = 1 - √6

So, the x-intercepts are x = 3, x = -3, x = 1 + √6, and x = 1 - √6. To express these as ordered pairs, we write them as (x, 0):

(3, 0)
(-3, 0)
(1 + √6, 0)
(1 - √6, 0)*

Determining Where f Crosses the Axis

Determining where the function crosses the x-axis involves understanding the behavior of the graph at each intercept. A function crosses the x-axis at an intercept if the multiplicity of the corresponding root is odd. In simpler terms, if the factor (x - r) appears an odd number of times in the factored form of the polynomial, the graph crosses the x-axis at x = r. If the factor appears an even number of times, the graph touches the x-axis but doesn't cross it (it bounces off).

In our case, the factored form of the polynomial is:

f(x) = (x - 3)(x + 3)(x - (1 + √6))(x - (1 - √6))

Each factor appears only once, meaning each root has a multiplicity of 1, which is odd. Therefore, the function crosses the x-axis at all four x-intercepts:

(3, 0)
(-3, 0)
(1 + √6, 0)
(1 - √6, 0)*

So, f(x) crosses the x-axis at all the x-intercepts we found. This is because each root has a multiplicity of one, indicating a change in the sign of f(x) as x passes through each intercept.

Expressing the Intercepts as Ordered Pairs

Expressing the intercepts as ordered pairs is the final step. We've already found the x-intercepts and confirmed that the function crosses the x-axis at each of these points. Now, we simply write them in the form (x, 0).

The x-intercepts are:

(3, 0): The function crosses the x-axis at x = 3.
(-3, 0): The function crosses the x-axis at x = -3.
(1 + √6, 0): The function crosses the x-axis at x = 1 + √6.
(1 - √6, 0): The function crosses the x-axis at x = 1 - √6.

So, these are the x-intercepts where the polynomial function f(x) = x^4 - 2x^3 - 18x^2 + 6x + 45 crosses the x-axis. We found these by factoring the polynomial, using the quadratic formula, and understanding the concept of multiplicity. Great job, guys! You've successfully navigated through this problem. Remember to always double-check your work and consider using graphing tools to visualize the functions you're working with. Keep up the great work!