Hard Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Write the degree of the polynomial 3x5 - 2x3 + x - 7.

Correct! The highest power of x is 5.

(2) Write the zero of the linear polynomial p(x) = 2x - 5. Zero of polynomial =

Perfect! The zero is x = 52.

(3) Find the value of k if x - 3 is a factor of x2 + kx - 15. k =

Excellent! k = 2.

(4) Write one example of a monomial, binomial, and trinomial.

Monomial: 5x22x+1

Binomial: 3x+73y2+5

Trinomial: x2−4x+3x3+2x

(5) Find the constant term of p(x) = x3 - 4x2 + 2x - 5.

Correct! The constant term is -5.

Short Answer Questions (2 Marks Each)

(1) If x + 2 is a factor of p(x) = x3 + 3x2 - 4x - 12, find all the factors of p(x). All factors: (x + 2), x+3x+5 and x−2x−3

(2) Using the Remainder Theorem, find the remainder when p(x) = x3 - 4x2 + 5x - 2 is divided by x - 2. Remainder =

Perfect! Remainder = 0, so x - 2 is a factor.

(3) If two zeros of p(x) = x3 - 3x2 - 4x + 12 are 2 and -2, find the third zero. Third zero =

Excellent! Third zero = 3.

(4) Find the value of p(1) + p(-1) if p(x) = x4 - 2x3 + x2 - x + 1. p(1) + p(-1) =

(5) Factorize x4 - 4x2 + 3. x2−3x−1x+1x−3x2−1x+1

Long Answer Questions (4 Marks Each)

(1) If x - 1 and x - 2 are factors of p(x) = x3 - 3x2 + ax + b, find a and b. a = , b =

Perfect! a = 2, b = 0.

(2) Divide x4 - 6x3 + 11x2 - 6x by x2 - 3x + 2 and find the quotient and remainder. Quotient: x2−3xx3+2x2+x+4 and Remainder:

(3) Solve the equation x3 - 4x2 - x + 4 = 0 by factorization. x = , x = , x =

(4) Verify that x + 1 is a factor of x4 + x3 - 7x2 - x + 6, and find the remaining factors.

x+1 isisn't a factor.

Other factors: x+1x−1x−2x+3x+1x−1x+2x+3

(5) The product of two factors of p(x) = x3 + ax2 + bx + c is x2 + 4x + 3 and the remaining factor is x - 2. Find a, b, and c. a = , b = and c =

Part B: Objective Questions (1 Mark Each)

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

(1) The degree of 4x5 - 2x4 + x is:

(a) 4 (b) 5 (c) 3 (d) 2

Correct! The highest power of x is 5.

(2) If x - 3 is a factor of x2 + kx - 15, then k =

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

Correct! k = 2.

(3) The remainder when x3 - 27 is divided by x - 3 is:

(a) 0 (b) 9 (c) 27 (d) 3

Correct! By Remainder Theorem: p(3) = 33 - 27 = 0.

(4) Which of these is NOT a polynomial?

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

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

(5) If the degree of p(x) is 0, then p(x) is a:

(a) Linear polynomial (b) Constant polynomial (c) Quadratic polynomial (d) Cubic polynomial

Correct! A polynomial of degree 0 is a constant polynomial.

(6) The number of zeros of x2 - 4 is:

(a) 1 (b) 2 (c) 3 (d) 0

Correct! x2 - 4 = (x - 2)(x + 2), so zeros are x = 2 and x = -2.

(7) The factor theorem is used to:

(a) Find factors of polynomials (b) Find degree of polynomials

(c) Find coefficient of terms (d) None of these

Correct! Factor theorem helps find factors of polynomials.

(8) The zero of 3x + 7 is:

(a) −73 (b) 73 (c) 3 (d) -3

Correct! 3x + 7 = 0 gives x = −73.

(9) The sum of zeros of x2 - 5x + 6 is:

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

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

(10) The product of zeros of 2x2 + 5x + 3 is:

(a) 32 (b) 3 (c) −32 (d) -3

Correct! For ax2 + bx + c, product of zeros = ca = 32.

Complex Polynomial Properties True or False

Determine whether these statements are True or False:

Polynomials and Factorisation - Hard Quiz

🎉 Outstanding Mastery! Advanced Polynomial Excellence Achieved:

You have successfully conquered the "Polynomials and Factorisation (Hard)" worksheet and mastered:

(1) Advanced Polynomial Classification: Understanding complex polynomial degrees and identifying monomials, binomials, and trinomials

(2) Zero and Root Analysis: Finding zeros of polynomials and understanding their relationship to factors

(3) Factor Theorem Mastery: Using the factor theorem to find unknown coefficients and verify factors

(4) Advanced Factorization: Factoring complex polynomials including quartic expressions and grouping methods

(5) Polynomial Division: Performing long division of polynomials and finding quotients and remainders

(6) Remainder Theorem Applications: Using the remainder theorem for complex polynomial divisions

(7) Systems of Equations: Solving for unknown coefficients using multiple factor conditions

(8) Sum and Product of Zeros: Understanding relationships between coefficients and zeros in quadratic polynomials

(9) Polynomial Evaluation: Calculating polynomial values for specific inputs efficiently

(10) Complex Factorization Techniques: Using substitution methods and advanced grouping for difficult expressions

(11) Equation Solving: Solving polynomial equations through systematic factorization

(12) Verification Methods: Confirming factors and solutions through substitution and calculation

(13) Polynomial Reconstruction: Building polynomials from given factor information

(14) Advanced Algebraic Manipulation: Working with complex polynomial expressions and transformations

(15) Multiple Factor Analysis: Finding all factors of higher-degree polynomials

(16) Coefficient Relationships: Understanding how polynomial structure relates to coefficients

(17) Mathematical Proof Techniques: Verifying polynomial properties through logical reasoning

(18) Strategic Problem-solving: Choosing optimal approaches for different types of polynomial problems

(19) Advanced Mathematical Communication: Expressing complex polynomial concepts clearly and precisely

Exceptional achievement! You've mastered advanced polynomial theory with sophisticated algebraic reasoning!