Possible Test Cases in Tester I am trying to write a human-readable tester for a functioning program.

There are four methods in the class Polynomial: public Polynomial(int[] coefficients), public int getDegree(), public int getCoefficient(int k), public long evaluate(int x).

Where coefficients is an immutable array of elements that represent a polynomial. For example, the polynomial 6x^3 + 4x + 3 would be represented by its coefficients as [3, 4, 0, 6]. The method getDegree() determines the highest order term that has a non-zero coefficient. For example, the degree would equal 3 in the previous example. getCoefficient(int k) returns the coefficient of element k. For example, in the previous example, if k = 2 the returned value is 0. The method evaluate(int x) evaluates the polynomial with the value of x. In the previous example, the value returned when x = 2 would be 59.

This is the tester I've written so far:

@Test public void testPolynomial()
    {
        int [] c1 = {0, 0, 0, 1};
        Polynomial p1 = new Polynomial(c1);
        c1[2] = 1;
        assertEquals("Degree error", 3, p1.getDegree());
        assertEquals("Polynomial must compute coefficients out of range", 0, p1.getCoefficient(-1)); 
        assertEquals("Polynomial must compute coefficients out of range", 0, p1.getCoefficient(4));
        assertEquals("Array is not immutable", 0, p1.getCoefficient(2));
        assertEquals("Coefficient error near boundary", 1, p1.getCoefficient(3)); 
        assertEquals("Coefficient error near boundary", 0, p1.getCoefficient(0));
        assertEquals("Coeffificent error", 0, p1.getCoefficient(1));
        assertEquals("Evaluation error", 0, p1.evaluate(0));
        assertEquals("Evaluation error of negative x", -27, p1.evaluate(-3));

        int[] c2 = {-1, 0, 0, 0};
        Polynomial p2 = new Polynomial(c2);
        c2[2] = 4;
        assertEquals("Degree error", 0, p2.getDegree());
        assertEquals("Polynomial must compute coefficients out of range", 0, p2.getCoefficient(-1));
        assertEquals("Polynomial must compute coefficients out of range", 0, p2.getCoefficient(4));
        assertEquals("Array is not immutable", 0, p2.getCoefficient(2));
        assertEquals("Coefficient error near noundary", 0, p2.getCoefficient(3));
        assertEquals("Coefficient error near noundary", -1, p2.getCoefficient(0));
        assertEquals("Evaluation error", -1, p2.evaluate(0));
        assertEquals("Evaluation error of negative x", -1, p2.evaluate(-3));

    }


Are there any other test cases that anyone can think of that I could include into my tester or that I'm missing?
I'm not going to address what else you could test, since it really depends what you're expecting your Polynomial class to do.  However, looking at your test, there's several things that could be improved.
Test One Thing
Having all of your assertions in the same test means that the test can fail for multiple different reasons, some of which are unrelated.  Consider modifying your tests so that they contain the complete AAA for what they are testing.  This makes it much clearer what the relationship between the data setup in the Arrange phase is, to the Assertions made.
As an example, your code current does this:

int [] c1 = {0, 0, 0, 1};
Polynomial p1 = new Polynomial(c1);
c1[2] = 1;
assertEquals(&quot;Degree error&quot;, 3, p1.getDegree());
assertEquals(&quot;Polynomial must compute coefficients out of range&quot;, 0, p1.getCoefficient(-1)); 
assertEquals(&quot;Polynomial must compute coefficients out of range&quot;, 0, p1.getCoefficient(4));
assertEquals(&quot;Array is not immutable&quot;, 0, p1.getCoefficient(2));


There's three unrelated assertions between the point where you modify your array (c1[2] = 1) and the assertion that it impacts 'Array it not immutable'.  This make far less obvious what the 'c1[2] = 1' is doing there.
Having multiple tests, where each one tests only a single thing also means that you don't need to put a message in each of your assertEquals calls.  The test name itself tells you what you're testing.
Duplicate tests
Sometimes you want to test the same thing in different scenarios.  Other times, it doesn't make as much sense.  I'm pretty sure that if the array is immutable for c1, it will also be immutable for c2, testing it again doesn't seem like it's really adding value.  If you do want to test the same thing, with different values, consider using parameterised tests.  Something like this:
@ParameterizedTest
@MethodSource(&quot;coefficients_outOfRangeCoefficient&quot;)
void getCoefficient_outOfRange_computesZero(int[] coefficients, int outOfRangeCoefficient) {
    Polynomial polynomial = new Polynomial(coefficients);
    assertEquals(0, polynomial.getCoefficient(outOfRangeCoefficient));
}

static Stream&lt;Arguments&gt; coefficients_outOfRangeCoefficient() {
    return Stream.of(
            arguments(new int[] {0,0,0,1}, -1),
            arguments(new int[] {0,0,0,1}, -4),
            arguments(new int[] {-1,0,0,0}, -1),
            arguments(new int[] {-1,0,0,0}, 4)
    );
}

Would allow the same test to be used across four different combinations of coefficients and get indexes.