Polynomials — Practice

Question 1

For p(x) = 5x^7 - 6x^5 + 7x - 6, write the coefficient of x^5, the degree of p(x), and the constant term.

Accepted Answer

coefficient of x^5 is -6; degree is 7; constant term is -6

Question 2

True or false, with a reason: the degree of the polynomial (square root of 2) x squared - 3x + 1 is the square root of 2.

Accepted Answer

False

Question 3

If p(t) = t cubed - 1, find p(1), p(-1), p(0), p(2) and p(-2).

Accepted Answer

0, -2, -1, 7, -9

Question 4

Find the zeroes of 4u squared + 8u.

Accepted Answer

0 and -2

Question 5

Find a quadratic polynomial whose zeroes have sum 4 and product 1.

Accepted Answer

x squared - 4x + 1

Question 6

Find the zeroes of p(x) = x squared + 5x + 6, and say how many crossings its graph makes with the X-axis.

Accepted Answer

-2 and -3; the parabola crosses the X-axis twice

Question 7

Without drawing, say how many zeroes each of these has and find them: (i) x squared - 6x + 9 (ii) x squared - 4x + 5.

Accepted Answer

(i) one zero, 3, repeated; (ii) no zeroes

Question 8

Find the zeroes of 6x squared - 3 - 7x and verify the relationship between the zeroes and the coefficients.

Accepted Answer

3/2 and -1/3

Question 9

Find a quadratic polynomial whose zeroes have sum the square root of 2 and product one third.

Accepted Answer

3x squared - 3(square root of 2)x + 1

Question 10

Divide p(x) = x cubed - 3x squared + 5x - 3 by g(x) = x squared - 2 and state the quotient and remainder.

Accepted Answer

quotient x - 3, remainder 7x - 9

Question 11

Verify that 1, -1 and 3 are the zeroes of the cubic polynomial x cubed - 3x squared - x + 3, and check all three relationships between the zeroes and the coefficients.

Accepted Answer

All three are zeroes; sum 3 = -b/a, sum of pairwise products -1 = c/a, product -3 = -d/a

Question 12

Obtain all the zeroes of 3x^4 + 6x^3 - 2x^2 - 10x - 5, given that two of them are plus and minus the square root of five thirds.

Accepted Answer

root(5/3), -root(5/3), -1 and -1

Question 13

On dividing x cubed - 3x squared + x + 2 by a polynomial g(x), the quotient and remainder were x - 2 and -2x + 4. Find g(x).

Accepted Answer

g(x) = x squared - x + 1

Question 14

Give an example of polynomials p(x), g(x), q(x) and r(x) satisfying the division algorithm with degree q(x) = degree r(x).

Accepted Answer

p(x) = x cubed + x squared + x + 1, g(x) = x squared - 1, q(x) = x + 1, r(x) = 2x + 2

Question 15

Find a cubic polynomial whose zeroes have sum 2, sum of products taken two at a time -7, and product -14.

Accepted Answer

x cubed - 2x squared - 7x + 14

Question 16

Two zeroes of x^4 - 6x^3 - 26x^2 + 138x - 35 are 2 plus and minus the square root of 3. Find the other zeroes.

Accepted Answer

7 and -5

Practice — Polynomials

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