x=a is a zero with even multiplicity. The function's sign does not change at x=a.

x=a is a zero with odd multiplicity. The function's sign changes at x=a.

The degree is related to the number of turning points (local maxima and minima). A polynomial of degree n can have at most n-1 turning points. Also, consider the end behavior.

Real zeros are the x-intercepts of the graph.

The end behavior indicates the sign of the leading coefficient and whether the degree is even or odd.

Symmetric about the y-axis.

Rotationally symmetric about the origin.

It suggests a zero with a higher multiplicity or a turning point.

Look at whether the graph is above (positive) or below (negative) the x-axis in each interval.

The value of the polynomial when x = 0, i.e., p(0).

1. Use the Rational Root Theorem to find potential rational roots. 2. Use synthetic division to test the potential roots. 3. Factor the polynomial completely. 4. Solve for the remaining zeros (may involve the quadratic formula).

1. Find f(-x). 2. If f(-x) = f(x), the function is even. 3. If f(-x) = -f(x), the function is odd. 4. If neither, the function is neither even nor odd.

1. Plot the x-intercepts (zeros). 2. Determine the end behavior based on the leading term. 3. Consider the behavior at each x-intercept based on multiplicity (cross or touch). 4. Sketch a smooth curve connecting the intercepts.

1. Find all real zeros. 2. Create a number line with the zeros. 3. Test a value in each interval. 4. Determine the sign of the polynomial in each interval.

1. Write the linear factors corresponding to each zero. 2. Raise each factor to the power of its multiplicity. 3. Multiply the factors together. 4. Multiply by a constant 'a' if another point on the polynomial is given.

1. Write down the coefficients of the polynomial and the potential root. 2. Bring down the first coefficient. 3. Multiply by the root and add to the next coefficient. 4. Repeat until the last coefficient. 5. If the remainder is 0, the root is a zero.

1. Group terms in pairs. 2. Factor out the greatest common factor from each pair. 3. If the binomial factors are the same, factor out the binomial. 4. Simplify.

1. Identify a, b, and c in the quadratic equation $ax^2 + bx + c = 0$. 2. Substitute into the quadratic formula: $x = frac{-b pm sqrt{b^2 - 4ac}}{2a}$. 3. Simplify. 4. If the discriminant ($b^2 - 4ac$) is negative, the roots are complex.

1. Identify the leading term ($ax^n$). 2. If n is even and a > 0, both ends go up. 3. If n is even and a < 0, both ends go down. 4. If n is odd and a > 0, left end goes down, right end goes up. 5. If n is odd and a < 0, left end goes up, right end goes down.

1. Calculate successive differences (first, second, third, etc.). 2. The degree is the number of times you need to take differences until you get a constant value.

Real zeros of a polynomial *p(x)* correspond to the x-intercepts of its graph, where the graph crosses the x-axis at the point *(a, 0)*.

If a zero *a* has an even multiplicity, the graph of *p(x)* touches the x-axis at *x = a* but does not cross it. The function's sign does not change at this zero.

Calculate first differences, then second, third, and so on, until you find a constant difference. The number of times you need to take differences will be the degree of the polynomial.

Even functions are symmetric about the y-axis and satisfy the property *f(-x) = f(x)*.

Odd functions are rotationally symmetric about the origin and satisfy the property *f(-x) = -f(x)*.

The graph crosses the x-axis at the zero. The sign of the function changes at this point.

If a polynomial has real coefficients, complex zeros always come in conjugate pairs. If *a + bi* is a zero, then *a - bi* is also a zero.

The function is even, meaning f(-x) = f(x). The graph is a mirror image across the y-axis.

The function is odd, meaning f(-x) = -f(x). The graph looks the same when rotated 180 degrees around the origin.

Odd multiplicity: graph crosses the x-axis. Even multiplicity: graph touches the x-axis and turns around.