Polynomials: Division, Remainder and Factor Theorems - Year 12 Maths

Question 1

Divide 2x^3 - x^2 + 3x - 2 by x - 1, expressing the answer as a quotient.

Accepted Answer

2x^2 + x - 1

Answer

2x^2 - x + 1

Answer

2x^2 + x + 1

Answer

2x^2 - x - 1

Question 2

Expand the polynomial (x - 2)³.

Accepted Answer

x³ - 6x² + 12x - 8

Answer

x³ + 6x² + 12x + 8

Answer

x³ - 8

Answer

x³ + 8

Question 3

If p(x) = 2x^5 - 3x^3 + 4x - 7 has a factor of (x - 2), what is the other factor in the form of (x + a)?

Accepted Answer

(x + 2)

Answer

(x + 1)

Answer

(x - 1)

Answer

(x - 2)

Question 4

Find the quotient when x^3 + 2x^2 - 5x + 6 is divided by x + 3.

Accepted Answer

x^2 - x + 2

Answer

x^2 - 5

Answer

x^2 + x - 2

Answer

x^2 + 5

Question 5

If (x - a) is a factor of p(x), which of the following is also a factor of p(x)?

Accepted Answer

(x + a)

Answer

(x - 1)

Answer

(x + 1)

Answer

(x² - a²)

Question 6

Simplify (rac{x^2 + 3x - 4}{x - 1}) in the form (x + a)?

Accepted Answer

$(x + 2)$

Answer

(x + 4)

Answer

$(x - 4)$

Answer

$(x - 2)$

Question 7

If (p(x)) is a polynomial of degree (n) and (q(x)) is a polynomial of degree (m), where (n > m), what is the maximum possible degree of the remainder when (p(x)) is divided by (q(x))?

Accepted Answer

(n - m - 1)

Answer

$(n - m)$

Answer

$(n + m)$

Answer

$(m - n)$

Question 8

Which of the following is NOT a factor of x³ - 8?

Accepted Answer

x² + 4

Answer

x - 2

Answer

x + 2

Answer

x² - 2

Question 9

If (x - 1) is a factor of the polynomial x⁴ + ax³ + bx² + cx - 6, what is the value of a + b + c?

Accepted Answer

11

Answer

-11

Answer

7

Answer

-7

Question 10

When the polynomial x⁴ + 2x³ - 5x² + kx + 6 is divided by (x + 2), the remainder is 10. Find the value of k.

Accepted Answer

-7

Answer

7

Answer

3

Answer

-3

Question 11

Which of the following polynomials is the result of (2x² + 3x - 1) multiplied by (x - 2)?

Accepted Answer

2x³ - x² - 7x + 2

Answer

2x³ - x² + 7x + 2

Answer

2x³ + x² - 7x + 2

Answer

2x³ + x² + 7x - 2

Question 12

What is the remainder when the polynomial x⁵ - 3x⁴ + 2x² - 5 is divided by (x + 1)?

Accepted Answer

-9

Answer

-1

Answer

1

Answer

9

Question 13

Which of the following is a factor of the polynomial 2x³ - 5x² + 3x - 2?

Accepted Answer

(2x - 1)

Answer

(x - 1)

Answer

(x + 1)

Answer

(x - 2)

Question 14

Use the Remainder Theorem to find the remainder when f(x) = x^4 - 2x^3 + 3x - 1 is divided by x - 1:

Accepted Answer

-3

Answer

-1

Answer

0

Answer

1

Question 15

Which of the following is a factor of x^4 - 1?

Accepted Answer

x - 1

Answer

x^3 + 1

Answer

x^2 + 1

Answer

x + 1

Question 16

Find the sum of all the coefficients of the polynomial (x - 1)^5:

Accepted Answer

32

Answer

0

Answer

16

Answer

-32

Question 17

The polynomial f(x) = x^3 - 2x^2 + kx has a remainder of 5 when divided by (x - 1). Find the value of k:

Accepted Answer

7

Answer

0

Answer

-5

Answer

5

Question 18

Which of the following is a consequence of Descartes' Rule of Signs?

Accepted Answer

A polynomial with an even number of sign changes in its coefficients has an even number of positive roots.

Answer

A polynomial with an odd number of sign changes in its coefficients has an odd number of negative roots.

Question 19

What is the degree of the polynomial f(x) = (x^2 - 1)(x + 2)^3?

Accepted Answer

5

Answer

3

Answer

6

Answer

8

Question 20

Which of the following is a possible rational root of the polynomial p(x) = x^3 - 2x^2 + 5x - 6?

Accepted Answer

2

Answer

-2/3

Answer

3

Question 21

When dividing the polynomial f(x) = x^3 - 2x^2 + 3x - 7 by the polynomial g(x) = x - 2, the remainder is:

Accepted Answer

3

Answer

1

Answer

0

Answer

2

Question 22

Which of the following is a factor of the polynomial x^3 - 1?

Accepted Answer

x - 1

Answer

x + 1

Answer

x^2 - 1

Answer

x^2 + 1

Question 23

If the polynomial f(x) leaves a remainder of 5 when divided by (x - 1) and a remainder of -2 when divided by (x + 2), find the remainder when f(x) is divided by (x^2 + x - 2).

Accepted Answer

1

Answer

0

Answer

2

Answer

-1

Question 24

The polynomial (x - 3)(x + 2) has:

Accepted Answer

Two real roots

Answer

One real root

Answer

No real roots

Answer

Three real roots

Question 25

Which of the following is the quotient when f(x) = x^4 - 2x^2 + 1 is divided by (x - 1)?

Accepted Answer

x^3 + x^2 + x + 1

Answer

x^3 - x^2 + x - 1

Answer

x^4 - 2x^2 + 1

Answer

x^4 + x^2 + 1

Question 26

The polynomial p(x) = x^3 + px^2 + q has a factor of (x - 2). If p = 1, find the value of q.

Accepted Answer

-3

Answer

2

Answer

-5

Answer

1

Question 27

Which of the following is the degree of the product of the polynomials f(x) = 2x - 1 and g(x) = x^2 + 1?

Accepted Answer

3

Answer

2

Answer

5

Answer

4

Question 28

The polynomial h(x) = x^3 - 4x + 3 can be expressed as the product of two linear and one quadratic factors. Which of the following could be one of the linear factors?

Accepted Answer

x - 1

Answer

x + 1

Answer

x + 3

Answer

x - 3

Question 29

The graph of the polynomial f(x) = (x - 1)(x + 2)(x - 3) has:

Accepted Answer

Three x-intercepts

Answer

Two x-intercepts

Answer

One x-intercept

Answer

No x-intercepts