Innings2
Create New Account

Sign in to Innings2

or

New Account     Reset Password     

Chapter 3: Polynomials

    Introduction

    What are Polynomials?

    Working With Polynomials

    Geometrical Meaning of the Zeroes of a Polynomial

    Relationship between Zeroes and Coefficients of a Polynomial

    Cubic Polynomials

    Division Alogorithm For Polynomials

    What We Have Discussed

    Enhanced Curriculum Support

    Easy Level Worksheet

    Moderate Level Worksheet

    Hard Level Worksheet

Powered by Innings 2

Reset Progress

This will delete your progress and chat data for all chapters in this course, and cannot be undone!

Glossary

Select one of the keywords on the left…

Chapter 3: Polynomials > Easy Level Worksheet

Easy Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Write the degree of the polynomial 3x2 + 5x - 7.

(2) Is x3 + 2x + 1 a polynomial? Justify your answer.

Correct! A polynomial has non-negative integer powers of the variable.

(3) Write one example of a linear polynomial.

Perfect! A linear polynomial has degree 1, like 2x + 3.

(4) Find the value of the polynomial p(x) = x2 + 3x + 2 at x = 1.

Excellent! p(1) = 12 + 3(1) + 2 = 1 + 3 + 2 = 6.

(5) What is the coefficient of x2 in 4x3 + 3x2 - x + 5?

Correct! The coefficient of x2 is 3.

Short Answer Questions (2 Marks Each)

Note: Answer each question with steps and explanation. Write down the answers on sheet and submit to the school subject teacher.

(1) Find the remainder when f(x) = x2 + 3x + 1 is divided by x - 1. Remainder =

Perfect! By remainder theorem, remainder = f(1) = 12 + 3(1) + 1 = 5.

(2) If one zero of a quadratic polynomial x2 - 5x + 6 is 2, find the other zero. Other zero =

Excellent! Sum of zeros = 5, so if one is 2, the other is 5 - 2 = 3.

(3) Verify whether x = 1 is a zero of the polynomial x2 - 3x + 2.

Correct! p(1) = 12 - 3(1) + 2 = 1 - 3 + 2 = 0, so x = 1 is a zero.

(4) Write the standard form of a quadratic polynomial whose zeros are 2 and 3.

Perfect! Using (x - α)(x - β): (x - 2)(x - 3) = x2 - 5x + 6.

(5) Find the value of the polynomial p(x) = 2x2 - 3x + 1 at x = -1.

Excellent! p(-1) = 2−12 - 3(-1) + 1 = 2 + 3 + 1 = 6.

Long Answer Questions (4 Marks Each)

Note: Answer each question with steps and explanation. Write down the answers on sheet and submit to the school subject teacher.

(1) If the zeros of the quadratic polynomial x2 + 2x + k are equal, find the value of k. k =

Perfect! For equal zeros, discriminant = 0: 4 - 4k = 0, so k = 1.

(2) Find the quotient and remainder when 2x2 + 3x + 1 is divided by x + 1. Quotient = and Remainder =

Excellent! Using long division: 2x2 + 3x + 1 = (x + 1)(2x + 1) + 0.

(3) Factorise the polynomial x2 + 5x + 6.

Perfect! We need two numbers that multiply to 6 and add to 5: 2 and 3.

(4) If the product of the zeros of a quadratic polynomial is 4 and their sum is -5, write the polynomial. x2 + +

Excellent! Using x2 - (sum)x + (product): x2 - (-5)x + 4 = x2 + 5x + 4.

(5) Find the zeros of the polynomial x2 - 7x + 10, and verify the relationship between the zeros and the coefficients. Zeros are and

Perfect! Factoring: (x - 2)(x - 5) = 0. Sum = 2 + 5 = 7 = -(-7), Product = 2 × 5 = 10.

Part B: Objective Questions (1 Mark Each)

Choose the correct answer and write the option (a/b/c/d)

(1) The degree of the polynomial 5x3 + 4x2 - x + 7 is

(a) 2 (b) 3 (c) 1 (d) 4

2
3
1
4

Correct! The highest power of x is 3, so the degree is 3.

(2) The value of p(x) = x2 + 2x + 1 at x = -1 is:

(a) 0 (b) 4 (c) -2 (d) 1

0
4
-2
1

Correct! p(-1) = −12 + 2(-1) + 1 = 1 - 2 + 1 = 0.

(3) The polynomial x2 + 7x + 10 factors into:

(a) (x + 2)(x + 5) (b) (x + 1)(x + 10) (c) (x - 2)(x - 5) (d) (x - 1)(x - 10)

(x + 2)(x + 5)
(x + 1)(x + 10)
(x - 2)(x - 5)
(x - 1)(x - 10)

Correct! We need two numbers that multiply to 10 and add to 7: 2 and 5.

(4) A zero of the polynomial x2 - 4x + 4 is:

(a) 2 (b) -2 (c) 4 (d) 0

2
-2
4
0

Correct! x2 - 4x + 4 = x−22, so x = 2 is a repeated zero.

(5) If α and β are the zeros of x2 - 5x + 6, then α + β =

(a) 5 (b) 6 (c) -5 (d) 1

5
6
-5
1

Correct! For ax2 + bx + c, sum of zeros = −ba = −−51 = 5.

(6) The remainder when x2 + 2x + 3 is divided by x - 1 is:

(a) 6 (b) 4 (c) 2 (d) 0

6
4
2
0

Correct! By remainder theorem: f(1) = 12 + 2(1) + 3 = 6.

(7) The polynomial x2 - 9 is an example of:

(a) Perfect square trinomial (b) Difference of squares (c) Cube of a binomial (d) Quadratic in linear form

Perfect square trinomial
Difference of squares
Cube of a binomial
Quadratic in linear form

Correct! x2 - 9 = x2 - 32 = (x + 3)(x - 3), which is difference of squares.

(8) A quadratic polynomial always has:

(a) At least one zero (b) Exactly one zero (c) At most two zeros (d) Three zeros

At least one zero
Exactly one zero
At most two zeros
Three zeros

Correct! A quadratic can have 0, 1, or 2 zeros, so at most two zeros.

(9) Which of the following is not a polynomial?

(a) x3 + 2x (b) x−1 + 2x (c) x2 + 1 (d) x4 + 5x + 1

x³ + 2x
x⁻¹ + 2x
x² + 1
x⁴ + 5x + 1

Correct! x−1 + 2x has a negative power, so it's not a polynomial.

(10) If the zeros of a polynomial are 1 and 4, the polynomial is:

(a) x2 + 5x + 4 (b) x2 - 5x + 4 (c) x2 - 5x + 6 (d) x2 - 5x + 3

x² + 5x + 4
x² - 5x + 4
x² - 5x + 6
x² - 5x + 3

Correct! (x - 1)(x - 4) = x2 - 5x + 4.
Degree
Factor theorem
Coefficient
Remainder theorem
(x - 2)(x - 3)
Zero of polynomial
x² - 5x + 6
Leading term
Polynomial Components
Polynomial Theorems
Polynomial Forms

Polynomials Challenge

Determine whether these statements about polynomials are True or False:

A polynomial can have non-negative integer powers only
The degree of 3x² + 5x³ - 2 is 2
A quadratic polynomial always has two real zeros
If p(a) = 0, then (x - a) is a factor of p(x)
The remainder when p(x) is divided by (x - a) is p(a)
x⁻² + 3x + 1 is a polynomial

Polynomials Quiz