College Algebra Questions and Problems With Solutions sample 6 : Polynomials

College algebra questions and problems on multiplying, dividing and finding polynomials are presented. The solutions are at the bottom of the page.

Question 1

Find the product of the polynomials P(x) = 2x^{ 2} - 3x and Q(x) = 3x^{ 2} + x - 5

Question 2

Find the quotient and the remainder when polynomial P(x) = 2x^{ 4} - x^{ 3} -3x^{ 2} + 7x - 13 is divided by Q(x) = x^{ 2} - 4

Question 3

Find the fourth degree polynomial P(x) whose graph is shown below.

Question 4

Find the remainder if 4 x^{ 200} + 5 x^{ 95} - 4 x^{ 21} + 2x - 6 is divided by x - 1

Question 5

Find the third degree polynomial with leading coefficient -2 and having zeros 1, -2 and 3.

Question 7

Find all zeros of the polynomial P(x) = 3x^{ 4} + 26 x^{ 3} + 74 x^{ 2} + 74 x + 15 knowing that P(x) has two rational zeros.

Question 7

Polynomial P(x) = x^{ 3} + x^{ 2} + c x + d has x - 2 as a factor and when divided by x - 1, the remainder is equal to 3. Find the constant c and d.

Question 8

P(x) is an even fourth degree polynomial function such that P(-2) = 0, P(x + 3) has a zero at x = 4 and P(0) = -9. Find P(x).

Question 9

The polynomial P(x) = 2 x^{ 4} + 3 x^{ 3} - 16 x^{ 2} - 17 x + 12 has zeros at x = 1/2 and x = -3. Find the other zeros.

Question 10

Factor the following polynomials:
a) f(x) = x^{ 3} - x^{ 2} - 4 x + 4
b) f(x) = 2 x^{ 2} - x^{ 3}
c) f(x) = x^{ 2} - 2 x^{ 4}
d) f(x) = -(x^{ 2} - 2 x - 3)^{ 2}
e) f(x) = x^{ 4} + 3 x^{ 3} + 3 x^{ 2} + x

Question 11

A polynomial of degree 4 has a negative leading coefficient and simple zeros (i.e. zeros of multiplicity 1) at x = 2, x = -2, x = 1 and x = - 1. Is the y intercept below the x axis or above the x axis?

Question 12

Match the function to the graph.
1) f(x) = (x + 1)(x - 1)^{ 2}(x + 2)^{ 2}
2) f(x) = -(x + 1)(x - 1)^{ 4}
3) f(x) = (x + 1)(x - 1)^{ 3}(x - 3)
4) f(x) = (x + 1)^{ 2}(x - 1)^{ 3}

Question 13

Give four different reasons why the graph below cannot possibly be the graph of p(x) = x^{ 4} - x^{ 2} + 1.

Question 14

The graph of polynomial f is shown below.
a) Is the degree of f even or odd?
b) Is the leading coefficient negative or positive?
c) Can you find the degree of f from the graph?Explain your answer.

Question 15

If the polynomial 3x^{3} + q x^{2} + p x + 7, where q and p are real numbers, is divided by x^{2} - 1, the remainder is 5x + 4. Find q and p.