To find the intervals of length 1 in which the polynomial is guaranteed to have a zero, we can use the Intermediate Value Theorem. According to the theorem, if a function is continuous on a closed interval [a, b], and if f(a) and f(b) have opposite signs, then there must be at least one zero in the interval (a, b).
First, let's plot the graph of the polynomial using a graphing utility:
h(x) = x^4 - 10x^2 + 2
The graphing utility will help us visualize the function and locate the zeros.
Upon graphing the function, we can see that it intersects the x-axis at multiple points. We will look for intervals of length 1 where the polynomial has zeros.
Let's divide the x-axis into intervals of length 1 and find the sign changes of the function within each interval. If there is a sign change within an interval, it means there is a zero within that interval.
Interval 1: [-4, -3]
h(-4) = (-4)^4 - 10(-4)^2 + 2 = 258 < 0
h(-3) = (-3)^4 - 10(-3)^2 + 2 = -4 > 0
Since there is a sign change within this interval, there is a zero within the interval [-4, -3].
Interval 2: [-3, -2]
h(-3) = -4 < 0
h(-2) = 24 > 0
Since there is a sign change within this interval, there is a zero within the interval [-3, -2].
Interval 3: [-2, -1]
h(-2) = 24 > 0
h(-1) = -9 < 0
Since there is a sign change within this interval, there is a zero within the interval [-2, -1].
Interval 4: [-1, 0]
h(-1) = -9 < 0
h(0) = 2 > 0
There is no sign change within this interval, so there may or may not be a zero within the interval. We will need to use a root finding method or the graphing utility's root feature to approximate the zero within this interval.
Interval 5: [0, 1]
h(0) = 2 > 0
h(1) = -7 < 0
Since there is a sign change within this interval, there is a zero within the interval [0, 1].
Using the root feature of a graphing utility, we can approximate the zeros of the function:
The zeros of the function h(x) = x^4 - 10x^2 + 2 are approximately:
x ≈ -3.162
x ≈ -1.725
x ≈ -0.618
x ≈ 3.162
Please note that these are approximate values and may not be exact.