Polynomial Functions: math test, Grade 11 – download the PDF

1 Operations with polynomials / 4 pts

Simplify each expression.

1. \((3x^2 - 4x + 1) - (x^2 + 2x - 6)\)
2. \((x - 3)(2x^2 + x - 5)\)
3. \((2x - 3)^2\)

2 Long division / 4 pts

Use long division to divide \(x^3 + 4x^2 - 3x + 12\) by \(x + 5\). Write the answer in the form \(p(x) = d(x)q(x) + r\).

3 Synthetic division and the Factor Theorem / 4 pts

Let \(p(x) = 2x^3 - 9x^2 + 4x + 15\).

1. Use synthetic division to divide \(p(x)\) by \(x - 3\).
2. What does the remainder tell you?
3. Factor \(p(x)\) completely and list its zeros.

4 End behavior and zeros / 3 pts

Let \(f(x) = -2x^5 + 3x^2 - 1\).

1. State the degree and the leading coefficient, and describe the end behavior.
2. How many complex zeros does \(f\) have, counting multiplicity? What is the greatest possible number of turning points?

5 Rational Root Theorem / 3 pts

Consider \(p(x) = 2x^3 + 3x^2 - 8x + 3\).

1. List the possible rational zeros.
2. Find all the zeros of \(p\).

6 Difference of cubes / 2 pts

Factor \(8x^3 - 125\) as far as possible using real numbers.