For the following quadratic function, determine the values of a so that it has no x-intercepts.

y = ax² - 18x + 13

Find the other zero of the quadratic equation if one of the zeros is 4 given that the vertex of the function is (8, 4).

For the following polynomial function, determine whether the real zero satisfies the given condition or not.

f(x) = x⁴ - 3x³ + 6x² - 12x + 10; no real zero greater than 3

Express that the given function has a real zero between the numbers given using the intermediate value theorem.
ƒ(x)=-4x³ + 9x² +2x -1; 0 and 2

Find the quotient of the following polynomials by using synthetic division.

(x⁴ - 1296) ÷ (x - 6)

Using long division, divide the following polynomials. (5x⁴ - 13x³ - 13x² + 22x - 2) ÷ (x - 3)

Consider the given function. Calculate its average rate of change from x₁ = 16 to x₂ = 25.

f(x) = 2√x

Identify the possible number of positive and negative real zeros of the following function by using the Descartes's Rule of Signs:

f(x) = 6x⁴ - x³ - 4x² - 2x + 9

Determine the end behavior of the graph of the following polynomial function using the Leading Coefficient Test and then use the end behavior to predict the graph:$f(x) = -x^6 + x^4$

Determine the end behavior of the graph of the following polynomial function using the Leading Coefficient Test: $f(x)=19x^{3}-9x^{2}+2x-8$

Determine whether the given polynomial $f(x) = 5x^{4}-9x^{2}-19$ has a real zero between $-2$ and 1 by using the Intermediate Value Theorem.