What is an example of a quintic polynomial function that has exactly four zeros?
If you mean four distinct zeros, any 5th-degree with a single repeated root.
x(x-1)(x+1)(x+168787)^2
is one.
All nth-degree polynomials have exactly n roots, some of which may be repeated.
Why did the polynomial go to the party? Because it wanted to make a "quint" of friends but got "four" the number of zeros!
Now, to answer your question, here's an example of a quintic polynomial function with exactly four zeros:
f(x) = (x - 1)(x + 2)(x - 3)(x + 4)(x - 5)
To find an example of a quintic polynomial function with exactly four zeros, we can start by considering a basic form of a quintic polynomial:
f(x) = (x - r1)(x - r2)(x - r3)(x - r4)(x - r5)
In this form, r1, r2, r3, r4, and r5 represent the zeros of the polynomial.
To satisfy the condition of having exactly four zeros, we can choose any four distinct values for r1, r2, r3, and r4. The fifth zero, r5, can be any value we want.
For example, let's choose the following values for r1, r2, r3, and r4:
r1 = 1
r2 = 2
r3 = 3
r4 = 4
Now, let's choose a value for r5:
r5 = 5
The polynomial function with these zeros becomes:
f(x) = (x - 1)(x - 2)(x - 3)(x - 4)(x - 5)
Expanding this expression, we get:
f(x) = (x^2 - 3x + 2)(x^2 - 7x + 12)(x - 5)
Multiplying these binomials, we get the quintic polynomial function:
f(x) = x^5 - 17x^4 + 107x^3 - 305x^2 + 360x - 120
So, an example of a quintic polynomial function with exactly four zeros is f(x) = x^5 - 17x^4 + 107x^3 - 305x^2 + 360x - 120, where the zeros are x = 1, x = 2, x = 3, and x = 4.
To find an example of a quintic polynomial function that has exactly four zeros, we need to create a polynomial of degree five and then determine the appropriate coefficients so that it has four zeros or solutions.
First, let's recall the general form of a quintic polynomial:
f(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f
Now, we know that a polynomial will have a zero at x = p if and only if (x - p) is a factor of the polynomial. Therefore, if we want a quintic polynomial with four zeros, we have (x - p1), (x - p2), (x - p3), and (x - p4) as factors.
Next, we can express the polynomial in factored form:
f(x) = (x - p1)(x - p2)(x - p3)(x - p4)g(x)
Here, g(x) represents an additional factor or any potential remaining factors.
Since we have four zeros, we have:
f(x) = (x - p1)(x - p2)(x - p3)(x - p4)
To make it a quintic polynomial, we need to multiply this expression by an additional factor g(x) of degree one:
f(x) = (x - p1)(x - p2)(x - p3)(x - p4)g(x)
This g(x) term ensures that the degree of the polynomial is five.
Therefore, to obtain an example, you can assign any values to p1, p2, p3, and p4. For instance, let's take:
p1 = 2, p2 = 1, p3 = -1, p4 = -3
The quintic polynomial function with exactly four zeros would be:
f(x) = (x - 2)(x - 1)(x + 1)(x + 3)g(x)
In this example, g(x) can be any linear factor.
This is just one example, and you can create many variations by choosing different values for p1, p2, p3, and p4, as long as you have four distinct zeros.