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

1 Is it a polynomial? ★★★

Decide whether each function is a polynomial function. Explain briefly.

1. \(f(x) = 5x^3 - \sqrt{2}\,x + 1\)
2. \(g(x) = \dfrac{4}{x} + x\)
3. \(h(x) = \sqrt{x} + 3\)
4. \(j(x) = \dfrac{x^2 + 1}{3}\)

2 Standard form and degree ★★★

Write \(f(x) = 9 - x + 7x^5 - 4x^3\) in standard form, then give its degree, leading coefficient and constant term.

3 Adding and subtracting ★★★

Simplify.

1. \((4x^3 - 2x^2 + x - 8) + (-x^3 + 5x^2 - 3x + 2)\)
2. \((6x^2 + x - 4) - (2x^2 - 3x + 9)\)

4 Multiplying binomials ★★★

Expand and simplify: (a) \((x + 5)(x - 3)\) (b) \((2x - 1)(3x + 4)\).

5 Evaluating with the Remainder Theorem ★★★

Let \(p(x) = x^3 - 2x^2 + 5x - 1\). Find \(p(2)\) and \(p(-1)\). What are the remainders when \(p(x)\) is divided by \(x - 2\) and by \(x + 1\)?

6 A first synthetic division ★★★

Use synthetic division to divide \(x^3 + 2x^2 - 5x - 6\) by \(x - 2\).

7 Cubes in two steps ★★★

Factor: (a) \(x^3 + 64\) (b) \(8x^3 - 1\).

8 Reading end behavior ★★★

Describe what the graph does as \(x \to -\infty\) and as \(x \to +\infty\) for each function.

1. \(f(x) = -2x^3 + x\)
2. \(g(x) = x^4 - 3x\)
3. \(h(x) = -x^2 + 5\)
4. \(j(x) = 3x^5 - x^2\)

9 Long division with a remainder ★★★

Divide \(x^3 - 6x^2 + 11x - 7\) by \(x - 2\) using long division, and write the result as \(p(x) = d(x)q(x) + r\).

10 Missing terms ★★★

Divide \(3x^3 + 2x - 5\) by \(x + 1\). Be careful with the missing \(x^2\) term.

11 Synthetic division, degree 4 ★★★

Use synthetic division to divide \(2x^4 - 5x^3 - x + 6\) by \(x - 2\), and then state \(p(2)\) for \(p(x) = 2x^4 - 5x^3 - x + 6\).

12 Finding a missing coefficient ★★★

The polynomial \(p(x) = x^3 + kx^2 - 4x - 12\) has \(x + 3\) as a factor. Find \(k\), then factor \(p(x)\) completely.

13 The packaging box ★★★

A box has width \(x\) inches, length \((x + 4)\) inches and height \((x - 1)\) inches. Its volume is \(42\) cubic inches. Find the dimensions of the box in inches and in centimeters (\(1\) in \(= 2.54\) cm).

14 Rational root candidates ★★★

List all possible rational zeros of \(p(x) = 3x^3 - x^2 + 8x - 10\). Then find the real zero and the other two zeros.

15 Sum and difference of cubes ★★★

Factor completely: (a) \(125x^3 + 27\) (b) \(54x^3 - 2\).

16 Reading a graph ★★★

The graph shows a cubic polynomial function \(k\) with a leading coefficient of \(1\).

1. Read the zeros from the graph.
2. Write \(k(x)\) in factored form and check it with the y-intercept \((0, 6)\).
3. Describe the end behavior.

17 Solving a cubic completely ★★★

Solve \(2x^3 - x^2 - 13x - 6 = 0\).

18 Building a polynomial from its zeros ★★★

Find a polynomial with real coefficients, leading coefficient \(1\) and the smallest possible degree, whose zeros include \(2\) and \(1 + 3i\). Write it in standard form.

19 Six zeros ★★★

Find all complex zeros of \(x^6 - 64\). Hint: write it as a difference of cubes, then factor each cubic.

20 Two conditions, two unknowns ★★★

Let \(p(x) = x^3 + ax^2 + bx - 6\). When \(p(x)\) is divided by \(x - 1\) the remainder is \(-8\), and \(x + 1\) is a factor of \(p(x)\). Find \(a\) and \(b\), then factor \(p(x)\) completely.

21 True or false? ★★★

Decide whether each statement is true or false and justify your answer.

1. Every real polynomial of odd degree has at least one real zero.
2. A real polynomial of degree \(4\) can have no real zeros.
3. A polynomial of degree \(5\) always has exactly \(5\) different real zeros.