A fourth degree polynomial equation is shown below. y = x^4 – 10x^2 + 9 Use this problem to complete the following questions. 1) Write the equation in factored form. 2) Use what you have learned to find the solutions of the polynomial equation: x^4 – 10x^2 + 9 = 0. Discuss the zeros of the graph of the polynomial equation and how many times the graph of the polynomial equation intersects the x-axis.
3. A fourth degree polynomial equation is shown below.
y = x^4 – 10x^2 + 9
Use this problem to complete the following questions.
1) Write the equation in factored form.
2) Use what you have learned to find the solutions of the polynomial equation: x^4 – 10x^2 + 9 = 0.
3)Discuss the zeros of the graph of the polynomial equation and how many times the graph of the polynomial equation intersects the x-axis.
The Correct Answer and Explanation is :
1) Writing the Equation in Factored Form:
We are given the fourth-degree polynomial equation:
[
y = x^4 – 10x^2 + 9
]
To write this equation in factored form, we first recognize that it can be treated as a quadratic equation in terms of (x^2). Let’s make a substitution to simplify:
Let ( u = x^2 ), so the equation becomes:
[
y = u^2 – 10u + 9
]
Now, we can factor this quadratic equation. We need to find two numbers that multiply to (9) (the constant term) and add up to (-10) (the coefficient of (u)).
The factors of 9 that add up to (-10) are (-1) and (-9). So, we can factor the quadratic equation as:
[
y = (u – 1)(u – 9)
]
Now, substitute back (u = x^2):
[
y = (x^2 – 1)(x^2 – 9)
]
Thus, the factored form of the equation is:
[
y = (x^2 – 1)(x^2 – 9)
]
2) Finding the Solutions of the Polynomial Equation:
We are tasked with finding the solutions to the equation:
[
x^4 – 10x^2 + 9 = 0
]
Using the factored form:
[
(x^2 – 1)(x^2 – 9) = 0
]
We can solve each factor separately.
- (x^2 – 1 = 0)
[
x^2 = 1 \quad \Rightarrow \quad x = \pm 1
]
- (x^2 – 9 = 0)
[
x^2 = 9 \quad \Rightarrow \quad x = \pm 3
]
Thus, the solutions to the equation are:
[
x = \pm 1, \pm 3
]
3) Discussing the Zeros of the Graph:
The zeros of the graph of the polynomial are the values of (x) that make (y = 0). From the solutions above, the zeros are (x = 1, -1, 3, -3).
These are the points where the graph intersects the x-axis. Since the equation is of the fourth degree, the graph will have up to four real solutions. In this case, we have four distinct real solutions, so the graph of the polynomial will intersect the x-axis at four points:
- At (x = 1)
- At (x = -1)
- At (x = 3)
- At (x = -3)
Each of these is a simple root, meaning the graph crosses the x-axis at each of these points. Therefore, the graph of the polynomial intersects the x-axis four times.
Explanation:
This problem demonstrates how we can factor a polynomial equation and find its solutions by treating it as a quadratic equation in terms of (x^2). By substituting (x^2) as a new variable, the fourth-degree polynomial reduces to a solvable quadratic form. After factoring, we find that the polynomial has four real solutions, which correspond to four points where the graph intersects the x-axis. The fact that the equation has four real roots indicates that the graph crosses the x-axis at four distinct points.