find the x-intercepts for P(x)= x^4-10x^2+9 and determine the behavior of the graph near the intercepts
To find the x-intercepts of the function P(x) = x^4 - 10x^2 + 9, we need to set the function equal to zero and solve for x.
Setting P(x) = 0, we have:
x^4 - 10x^2 + 9 = 0
Now, let's factor the equation:
(x^2 - 9)(x^2 - 1) = 0
Now, we can set each factor equal to zero and solve for x:
For x^2 - 9 = 0:
x^2 = 9
Taking the square root of both sides:
x = ±3
For x^2 - 1 = 0:
x^2 = 1
Taking the square root of both sides:
x = ±1
Therefore, the x-intercepts of P(x) = x^4 - 10x^2 + 9 are x = -3, -1, 1, and 3.
Now, let's determine the behavior of the graph near each of these intercepts.
1. x = -3:
As we approach x = -3 from the left, the graph of P(x) increases without bound. As we approach x = -3 from the right, the graph decreases without bound.
2. x = -1:
As we approach x = -1 from the left, the graph of P(x) decreases without bound. As we approach x = -1 from the right, the graph increases without bound.
3. x = 1:
As we approach x = 1 from the left, the graph of P(x) increases without bound. As we approach x = 1 from the right, the graph decreases without bound.
4. x = 3:
As we approach x = 3 from the left, the graph of P(x) decreases without bound. As we approach x = 3 from the right, the graph increases without bound.
In summary, near each x-intercept:
- At x = -3 and x = 1, the graph approaches positive infinity from both sides.
- At x = -1 and x = 3, the graph approaches negative infinity from both sides.
To find the x-intercepts of a function, we need to solve the equation P(x) = 0. In this case, our function is P(x) = x^4 - 10x^2 + 9.
Step 1: Set P(x) equal to zero:
x^4 - 10x^2 + 9 = 0
Step 2: Factor the equation, if possible:
Unfortunately, this equation cannot be factored further using simple factorization techniques.
Step 3: Solve the equation:
To solve this equation, we can use the fact that a quadratic equation in the form ax^2 + bx + c = 0 can be solved using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, the equation is in the form of a quadratic equation with x^2 as the variable. Let's substitute x^2 with a new variable, say u, and solve for u.
u^2 - 10u + 9 = 0
Using the quadratic formula, we have:
u = (-(-10) ± √((-10)^2 - 4(1)(9))) / (2(1))
u = (10 ± √(100 - 36)) / 2
u = (10 ± √64) / 2
u = (10 ± 8) / 2
u = 9 or u = 1
Now we substitute the values of u back into the equation x^2 = u:
x^2 = 9
or
x^2 = 1
Taking the square root of both sides:
x = ±3
or
x = ±1
So, the x-intercepts of the function P(x) = x^4 - 10x^2 + 9 are x = -3, -1, 1, and 3.
To determine the behavior of the graph near the x-intercepts, we can look at the signs of the function values on either side of each intercept.
-3: Substitute x = -4 (a value smaller than -3) and x = -2 (a value greater than -3) into P(x):
P(-4) = (-4)^4 - 10(-4)^2 + 9 = 337
P(-2) = (-2)^4 - 10(-2)^2 + 9 = 1
Since the function values on both sides of x = -3 have different signs (one positive and one negative), we conclude that the graph of the function crosses the x-axis at x = -3.
-1: Substitute x = -2 (a value smaller than -1) and x = 0 (a value greater than -1) into P(x):
P(-2) = (-2)^4 - 10(-2)^2 + 9 = 1
P(0) = 0^4 - 10(0)^2 + 9 = 9
Since the function values on both sides of x = -1 have different signs, we conclude that the graph of the function crosses the x-axis at x = -1.
1: Substitute x = 0 (a value smaller than 1) and x = 2 (a value greater than 1) into P(x):
P(0) = 0^4 - 10(0)^2 + 9 = 9
P(2) = 2^4 - 10(2)^2 + 9 = 1
Since the function values on both sides of x = 1 have different signs, we conclude that the graph of the function crosses the x-axis at x = 1.
3: Substitute x = 2 (a value smaller than 3) and x = 4 (a value greater than 3) into P(x):
P(2) = 2^4 - 10(2)^2 + 9 = 1
P(4) = 4^4 - 10(4)^2 + 9 = 337
Since the function values on both sides of x = 3 have different signs, we conclude that the graph of the function crosses the x-axis at x = 3.
Based on this analysis, the behavior of the graph near each x-intercept is that it crosses the x-axis at each point.