Hard Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Find the degree of the polynomial x4 − 2x3 + x2 − x + 1. Degree =

Perfect! The degree is the highest power of the variable.

(2) Write the relationship between the zeros and coefficients of the quadratic polynomial ax2 + bx + c.

Sum of zeros =

Product of zeros =

Excellent! These are the fundamental relationships for quadratic polynomials.

(3) What is the value of p(2), if p(x) = x3 − x? p(2) =

Correct! p(2) = 23 − 2 = 8 − 2 = 6.

(4) Can x−2 + 3x be a polynomial? Why or why not? NoYes

Correct! Polynomials cannot have negative powers of the variable.

(5) What is the remainder when x2 − 2x + 1 is divided by x − 1? Remainder =

Perfect! By remainder theorem, remainder = p(1) = 1 − 2 + 1 = 0.

Short Answer Questions (2 Marks Each)

(1) Factorise x2 − 2x − 15 and find its zeros.

Zeros: x = , x =

Excellent! Always verify by expanding back to original form.

(2) Write a quadratic polynomial whose sum and product of zeros are 4 and −5 respectively.

Polynomial: x2 + +

Perfect! Remember the standard form x2 − Sx + P.

(3) Verify that the zeros of the polynomial x2 + x − 6 satisfy the relationship: Sum = −ba, Product = ca.

Zeros: x = , x =

Sum = = −11

Product = = −61

Excellent verification! The relationships hold true.

(4) Divide x3 − 3x2 + 5x − 3 by x − 1 and write the quotient and remainder.

Quotient: x2−2x+3x2+5x−3x2−4x+4

Remainder:

Perfect! Since remainder is 0, (x − 1) is a factor.

(5) One zero of a polynomial is 2. If the polynomial is x2−k+2x+2k, find the value of k.

k =

Well done! You got it right.

Long Answer Questions (4 Marks Each)

(1) If α and β are the zeros of the polynomial f(x) = x2 − 7x + 10, (i) Find α, β (ii) Verify the relationship between zeros and coefficients.

Zeros: α = , β =

Sum verification: α + β = = −−71

Product verification: αβ = = 101

Excellent! Both relationships are verified.

(2) Find the remainder and quotient when f(x) = x3 − 6x2 + 11x − 6 is divided by x − 1. Also, factorise the polynomial completely.

Remainder =

Quotient: x2−5x+6x2−3x−6x2+6x+8

Factorisation: x−1x−2x−3x−1x+2x−3x+1x−2x+3

Perfect! All factors found using repeated division.

(3) Construct a quadratic polynomial whose zeros are reciprocal of the zeros of 2x2 + 5x + 3.

Original zeros: α = , β =

Reciprocal zeros: 1α = , 1β =

Sum of reciprocals =

Product of reciprocals =

Required polynomial: 3x22x2 + +

Excellent! Always clear fractions in the final answer.

(4) The product of two of the zeros of a cubic polynomial f(x) = ax3 + bx2 + cx + d is 4 and the sum of all the three zeros is −3. If a = 1, b = 3, and c = −7, find the value of d.

d =

Excellent application of Vieta's formulas!

(5) If the zeros of the quadratic polynomial f(x) = x2 + ax + b are equal in magnitude but opposite in sign, then prove that a = 0 and write the value of the product of its zeros.

Product of zeros:

The product is ≤ 0negative or zero

Perfect proof! When zeros are opposites, their sum is always zero.

Part B: Objective Questions (1 Mark Each)

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

(1) If one zero of the polynomial x2 + px + 12 is 3, then the value of p is:

(a) −7 (b) −3 (c) −4 (d) −6

Correct! If x = 3 is a zero: 9 + 3p + 12 = 0, so p = −7.

(2) The quadratic polynomial whose zeros are 2 and −2 is

(a) x2 − 4 (b) x2 + 4 (c) x2 + 2x + 4 (d) x2 − 2x + 4

Correct! Sum = 0, Product = −4, so polynomial is x2 − 0x + (−4) = x2 − 4.

(3) The sum and product of the zeros of x2 − 5x + 6 are

(a) 5, 6 (b) −5, 6 (c) 5, −6 (d) −5, −6

Correct! Sum = −−51 = 5, Product = 61 = 6.

(4) If the remainder is 0 when a polynomial is divided by x − a, then

(a) x = a is not a zero (b) x = a is a zero (c) The degree of the polynomial is 1 (d) The polynomial is linear

Correct! This is the factor theorem: if remainder is 0, then x = a is a zero.

(5) The zero of the polynomial f(x) = x2 − 9 is

(a) 3 only (b) −3 only (c) 3 and −3 (d) 0

Correct! x2 − 9 = (x − 3)(x + 3), so zeros are 3 and −3.

(6) If f(x) = x3 − 3x2 + x + 1, then the value of f(2) is

(a) 3 (b) 5 (c) -1 (d) 0

Correct!

(7) The quadratic polynomial whose zeros are −1 and −2 is

(a) x2 + 3x + 2 (b) x2 − 3x + 2 (c) x2 + x − 2 (d) x2 − x − 2

Correct! Sum = −1 + (−2) = −3, Product = (−1)(−2) = 2, so x2 − (−3)x + 2 = x2 + 3x + 2.

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

(a) x3 + 2x2 + 1 (b) x2 − 1x + 5 (c) 4x2 + 3x (d) x + 7

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

(9) A polynomial of degree 2 has

(a) At most 1 zero (b) At most 2 zeros (c) At most 3 zeros (d) Infinitely many zeros

Correct! A polynomial of degree n has at most n zeros.

(10) If the sum of zeros is 0 and product is −4, the polynomial is

(a) x2 − 4 (b) x2 + 4 (c) x2 + 2x − 4 (d) x2 − 2x + 4

Correct! Using x2 − (sum)x + (product): x2 − 0x + (−4) = x2 − 4.

Polynomials Challenge

Determine whether these statements about polynomials are True or False:

Polynomials Quiz

🎉 You Did It! What You've Learned:

By completing this worksheet, you now have a solid understanding of:

(1) Polynomial Identification: Recognizing valid polynomials and their degrees

(2) Zero-Coefficient Relationships: Understanding sum and product formulas for quadratic polynomials

(3) Factorization: Breaking down polynomials into linear factors

(4) Remainder and Factor Theorems: Using these to find remainders and factors efficiently

(5) Polynomial Division: Dividing polynomials and finding quotients and remainders

(6) Constructing Polynomials: Building polynomials from given zero conditions

(7) Vieta's Formulas: Advanced relationships between zeros and coefficients

(8) Problem Solving: Applying polynomial concepts to solve complex mathematical problems

Excellent work mastering advanced polynomial concepts and their applications!