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SecA

1. The roots of the equation x2 + x - p (p+1) = 0, where p is a constant, are

2. The sum of the zeroes of the polynomial 2x2-4x+6 is

3. The degree of the polynomial 4x3+2x2-5x+7x is:

4. If one root of the quadratic polynomial x2+3x + k is 2, the value of k is:

5. Show that 𝑥 = −3 is a solution of 2x2 + 5x - 3 = 0

6. Write the polynomial, the product and sum of whose zeroes are −92 and −32 respectively.

SecB

1. Determine the polynomial whose zeroes are α and β given α+β = 3 and αβ=2

2.Find the zeroes of the polynomial x2- 5x + 6

3. What is the remainder when x3 -3x2+ x -5 is divided by x−1?

4. If two zeroes of the polynomial x2 -4x2 - 3x + 12 are √3 and -√3, then find its third zero

5. If the polynomial 6x4 + 8x3 + 17x2 + 21x + 7 is divided by another polynomial 3x2 + 4x + 1, the remainder comes out to be (ax + b), find a and b.

6. Write a quadratic polynomial, sum of whose zeroes is 2√3 and their product is 2.

SecC

1. Find the zeroes of the polynomial 2x2 -8x + 6 and verify the relationship between the zeroes and the coefficients.

2. If the polynomial x3-6x2 + 11x -6 is divided by x−2, what is the quotient?

3. Find the zeroes of the quadratic polynomial x2 + 5x + 6 and verify the relationship between the zeroes and the coefficients.

4. Find all the zeroes of the polynomial 3x4 + 6x3 - 2x2 - 10x -5 if two of its zeroes are 53 and −53

5. If two zeroes of polynomial x4 + 3x3 - 20x2 - 6x + 36 are √2 and -√2, find the other zeroes of the polynomial.

SecD

1. Find a quadratic polynomial whose zeroes are 5 and -3. Also, verify the relationship between the zeroes and the coefficients.

2. Divide 4x3 + 2x2 + 5x - 6 by 2x2 + 1 + 3x and verify the division algorithm.

3. Given that x – √5 is a factor of the polynomial x3 – 3√5 x2 – 5x + 15√5, find all the zeroes of the polynomial.

4. If α and β are the zeroes of the polynomial p(x) = 2x2 + 5x + k, satisfying the relation, α2 + β2 + αβ = 214 then find the value of k.

5. What must be subtracted from p(x) = 8x4 + 14x3 – 2x2 + 8x – 12 so that 4x2 + 3x – 2 is factor of p(x)?

Problem 1

Situation:

A farmer wants to divide his rectangular land into smaller plots. The length of the land is l meters, and the width is w meters. He wants to divide the length and width into smaller plots each increasing by x meters.

1. Find the area of the land.

2. Calculate the area when l=50 meters, w= 30 meters, and x= 5 meters.

Problem 2

Situation:

The population of a town is modeled by a polynomial function P(t) = t3 + 2 t2 + t + 1 where t is the time in years.

1. Find the population after 3 years.

Problem 3

Situation:

A factory produces a certain number of goods per day, and the daily production cost is represented by a polynomial function of the number of goods produced. The cost function is given P(x) = 2x2 - 5x + 10, where x represents the number of goods.

1. If the factory produces 10 goods in a day, calculate the total production cost. What is the cost when 15 goods are produced?

Problem 4

Situation:

You are designing a garden with a square-shaped lawn, and the remaining area is to be covered with flowers. The total area of the garden is represented by the polynomial P(x) = x2 + 6x + 9, where x is the length of one side of the square.

1. Find the dimensions of the square lawn if the total area of the garden is 49 square meters. How much area is available for flowers?

Q1

1. Find the Quadratic Polynomial whose sum and product of zeroes are 2+ 1, 12+ 1.

2. Create a polynomial equation of degree 4 with roots 2, -3, 4, and -5. Factorize the polynomial completely and verify the roots.

3. Can x – 1 be the remainder on division of a polynomial p (x) by 2x + 3? Justify your answer.

Q2

1. On dividing the polynomial 4x4 - 5x3-39x2-46x-2 by the polynomial g(x) the quotient is x2-3x-5 and the remainder is -5x+8. Find the polynomial g(x).

2. Given two polynomials p(x) = x3 - 3x + 2 and p(x) = x3 + x2 - 2x - 1, determine which polynomial has a greater sum of roots. Explain the reasoning behind your answer.

3. If all the zeros of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign. Is the statement true or false? Justify your answer

Q3

1. If the squared difference of the zeroes of the quadratic polynomial x2+ px +45 is equal to 144,find the value of p.

2. For the polynomial p(x) = 2x3 - 5x2 + kx + 7, if one of its roots is the reciprocal of another, find the value of k.

3. Given that the zeroes of the cubic polynomial x3- 6x2 + 3x + 10 are of the form a, a + b, a + 2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial.

Q4

1. Given that x = 2 and x = -3 are roots of the polynomial 2x3 + 3x2 - 11x - 6, find the third root without completely factorizing the polynomial.

2. Construct a real-life problem that can be modeled using a quadratic polynomial, solve the problem, and interpret the solution in context.

3. Given that x - √5 is a factor of the cubic polynomial x3 - 3 √5 x2 + 13x - 3 √5 find all the zeroes of the polynomial.

Questions

1. A quadratic polynomial, whose zeroes are –3 and 4, is

2. Find a quadratic polynomial, the sum and product of whose zeroes are √2 and -32 respectively. Also find its zeroes.

3. If the remainder on division of x3+2x2+ kx +3 by x – 3 is 21, find the quotient and the value of k. Hence, find the zeroes of the cubic polynomial x3+2x2+ kx-18.

4. Find the zeroes of the polynomial x2+16x -2, and verify the relation between the coefficients and the zeroes of the polynomial.

5. Can x – 1 be the remainder on division of a polynomial p (x) by 2x + 3? Justify your answer.

6. Is the following statement True or False? Justify your answer. If the zeroes of a quadratic polynomial ax2+bx+c are both negative, then a, b and c all have the same sign.

7. Find the zeroes of the polynomial x2 + 16x - 2, and verify the relation between the coefficients and the zeroes of the polynomial.

8. Find a quadratic polynomial, the sum and product of whose zeroes are √2 and −32, respectively. Also find its zeroes.

9. Given that two of the zeroes of the cubic polynomial ax3 + bx2 + cx + d are 0, the third zero is

10. A quadratic polynomial, whose zeroes are –3 and 4, is

Question 1

The natural shape of a banana can show the curve of a quadratic polynomial, which is in the form of p(x) = ax2+bx+c, where a,b and c are real numbers, a ≠ 0. By analysing the given figure, we can see that a quadratic polynomial is able to describe the shape of a banana quite accurately, with a = 0.1, b = 0 anc c = 0 Therefore, the polynomial is p(x) = 0.1x2

Based on your understanding of the above case study,answer all the five questions below:

(1) The number of zeroes in the polynomial for the shape of the banana is?

(2) If the curve of the banana is represented by the polynomial then the zeroes are?

(3) If the representation of curve of banana whose one zero is 4 and the sum of the zeroes is 0, the quadratic polynomial is? x2-16x-12

(4) Each zero of the polynomial is decreased by 2. The resulting polynomial is P(x)= x2 -2x +1 then

Answer: The new zeroes are and (both zeroes are equal).

(5) If the curve representing the polynomial P(x) = (m2+n2)x2 - 2 (mp+nq) x + p2 + q2 has equal zeroes then,

Answer: mp+nq2 = (m2+n2)(m2+p2) (p2+q2)(n2+q2)

Question 2

To transmit a signal, a controller sends it through the horn, and the dish focuses the signal into a relatively narrow beam. When the signal reaches the viewer's house, it is captured by the satellite dish. A satellite dish is just a special kind of antenna designed to focus on a specific broadcast source. The standard dish consists of a parabolic (bowl-shaped) surface and a central feed horn. To transmit a signal, a controller sends it through the horn, and the dish focuses the signal into a relatively narrow beam.

Based on the above figure, answer the following questions:

1. The zeroes of the quadratic polynomial representing the curve of dish are then the polynomial is?

Answer: P(x)ax2x2

2. If one of the zeroes of a quadratic polynomial representing the curve of the dish of the form P(x)= x2 + ax + b is the negative of the other, then which of the following statement is correct?

Answer: The sum of the zeroes is zero, so a =

3. The number of polynomials having 3 and 7 as zeroes are? Infinitely manythree

4. If α and β are the zeroes of the polynomial P(x)= x2 - px + 36 and α2 + β2 = 9 then the value of p is?

Answer: p = 9 or -9-9

5. If the polynomial representing the curve is P(x)= x2 - x6 - 15 then, One of the factors of the polynomial is

Answer: One factor will be of the form (x - r), where r is one of the roots determined by solving the quadraticpolynomial equation.