Moderate Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Write the zero of the linear polynomial f(x) = 2x - 6. x =

Perfect! For linear polynomials, set f(x) = 0 and solve for x.

(2) Find the remainder when x2 + 2x + 1 is divided by x + 1.

Remainder:

Excellent! Remainder theorem: when p(x) is divided by (x-a), remainder = p(a).

(3) What is the value of the polynomial p(x) = x2 - x + 1 at x = 0?

Correct! Simply substitute the given value into the polynomial.

(4) What do you call a polynomial of degree 1? polynomial

Perfect! Degree determines the type: 1 = linear, 2 = quadratic, 3 = cubic.

(5) If the product of the zeros of a quadratic polynomial is 4, write the value of ca. ca =

Excellent! This is a fundamental relationship for quadratic polynomials.

Short Answer Questions (2 Marks Each)

(1) Verify whether x = 1 and x = -1 are the zeros of the polynomial p(x) = x2 - 1. YesNo

Perfect! Both values make the polynomial equal to zero.

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

Excellent use of the sum of zeros formula!

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

Perfect! Careful with signs when substituting negative values.

(4) Construct a quadratic polynomial whose zeros are 3 and -5.

Final polynomial: p(x) = x2+2x−152x2+5x−20

Excellent! You can construct polynomials from their zeros.

(5) Use the factor theorem to show that x = -1 is a factor of the polynomial f(x) = x3 + x2 - x - 1. (x + 1)(x - 1) is a factor

Perfect application of the factor theorem!

Long Answer Questions (4 Marks Each)

(1) Divide the polynomial x3+3x2−x−3 by x + 1 and verify the division algorithm. Quotient: x2+2x−3x2−2x+3

Perfect long division! The division algorithm is verified.

(2) Find the quadratic polynomial whose sum and product of the zeros are -3 and 2 respectively. Quadratic polynomial: x2+3x+2x2−3x−2

Excellent! This formula constructs polynomials from their zero relationships.

(3) If α and β are the zeros of x2+2x−15, verify the relationship between the coefficients and the zeros.

Sum of zeros: α + β =

Product of zeros: αβ =

Perfect verification of the fundamental relationships!

(4) Factorize the polynomial x3−6x2+11x−6 completely and find its zeros.

Zeros: x = , x = , x =

Excellent systematic factorization!

Part B: Objective Questions (1 Mark Each)

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

(1) Which of the following is a cubic polynomial?

(a) x2+3x+2 (b) x3+4x2+x+5 (c) 2x + 1 (d) x4−x2

Correct! A cubic polynomial has degree 3 (highest power is 3).

(2) If x = 1 is a zero of p(x) = x2−2x+1, then the value of p(1) is:

(a) 0 (b) 1 (c) –1 (d) 2

Correct! If x = 1 is a zero, then p(1) = 0 by definition.

(3) The number of zeros of a cubic polynomial is:

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

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

(4) The polynomial x2−9 is equal to:

(a) x+32 (b) x−32 (c) (x + 3)(x - 3) (d) None

Correct! This is the difference of squares: a2 - b2 = (a + b)(a - b).

(5) If α and β are the zeros of the polynomial x2+4x+3, then α + β =:

(a) 3 (b) –3 (c) –4 (d) 4

Correct! Sum of zeros = −ba = −41 = -4.

(6) A polynomial p(x) is divisible by x - 2, then p(2) =:

(a) 2 (b) 0 (c) –2 (d) Not defined

Correct! By the factor theorem, if (x-a) is a factor, then p(a) = 0.

(7) What is the zero of the polynomial f(x) = 5?

(a) 0 (b) 1 (c) 5 (d) No zero

Correct! A constant polynomial (non-zero) has no zeros.

(8) A quadratic polynomial with both zeros equal is called:

(a) Perfect square (b) Repeated root (c) Double root (d) All the above

Correct! All these terms describe the same concept.

(9) The value of the polynomial x2+2x+1 at x = –1 is:

(a) 0 (b) 1 (c) –1 (d) 2

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

(10) Which of the following represents the standard form of a quadratic polynomial?

(a) ax2+bx+c (b) ax + b (c) ax3+bx2+cx+d (d) None

Correct! This is the standard form of a quadratic polynomial.

Polynomial Properties Challenge

Determine whether these statements about polynomials are True or False:

Polynomials Quiz