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Worksheet
Explore the world of Polynomials, understanding their types, degrees, and operations to solve algebraic expressions and equations effectively.
Polynomials - Practice Worksheet
Strengthen your foundation with key concepts and basic applications.
This worksheet covers essential long-answer questions to help you build confidence in Polynomials from Mathematics for Class X (Mathematics).
Questions
Define a polynomial and explain its types with examples.
Recall the definitions and examples provided in the introduction section of the chapter.
Find the zeroes of the polynomial x^2 - 5x + 6 and verify the relationship between the zeroes and the coefficients.
Use the factorization method to find the zeroes and recall the formulas for sum and product of zeroes.
Explain the geometrical meaning of the zeroes of a quadratic polynomial.
Think about the graph of a quadratic polynomial and how it relates to its zeroes.
If the sum and product of the zeroes of a quadratic polynomial are 4 and 1 respectively, find the polynomial.
Use the standard form of a quadratic polynomial in terms of the sum and product of its zeroes.
What is the division algorithm for polynomials? Explain with an example.
Recall the steps of polynomial division and the condition on the remainder.
Find all the zeroes of the polynomial x^3 - 4x^2 - 7x + 10, if two of its zeroes are 1 and -2.
Use the given zeroes to factorize the polynomial and find the remaining zero.
Explain why the polynomial x^2 + 1 has no real zeroes.
Consider the properties of real numbers and the definition of zeroes of a polynomial.
If the zeroes of the polynomial x^3 - 3x^2 + x + 1 are a - b, a, and a + b, find the values of a and b.
Use the sum and product of zeroes formulas for a cubic polynomial.
Verify that 2, -1, and -1/2 are the zeroes of the cubic polynomial 2x^3 - x^2 - 5x - 2.
Substitute each candidate zero into the polynomial and check if the result is zero.
Find a quadratic polynomial whose zeroes are the reciprocals of the zeroes of the polynomial x^2 - 4x + 3.
First find the zeroes of the given polynomial, then find their reciprocals, and use the sum and product to form the new polynomial.
Explain the significance of the remainder theorem with an example.
Recall the statement of the remainder theorem and how it simplifies polynomial evaluation.
Polynomials - Mastery Worksheet
Advance your understanding through integrative and tricky questions.
This worksheet challenges you with deeper, multi-concept long-answer questions from Polynomials to prepare for higher-weightage questions in Class X Mathematics.
Questions
Explain the difference between a linear polynomial and a quadratic polynomial with examples. Also, discuss their graphical representations.
Consider the degree and the general form of each polynomial type. Think about how their graphs differ in shape and intersection points with the x-axis.
Find the zeroes of the polynomial x² - 5x + 6 and verify the relationship between the zeroes and the coefficients.
Factor the quadratic polynomial to find its zeroes. Use the formulas for sum and product of zeroes in terms of coefficients.
If one zero of the quadratic polynomial x² + 3x + k is 2, find the value of k and the other zero.
Use the given zero to find k. Then, factor the polynomial to find the other zero.
Compare the number of zeroes a linear polynomial and a cubic polynomial can have. Justify your answer with examples.
Recall that the maximum number of zeroes a polynomial can have is equal to its degree.
Given that the sum and product of the zeroes of a quadratic polynomial are -3 and 2 respectively, find the polynomial.
Use the standard form of a quadratic polynomial in terms of the sum and product of its zeroes.
Explain why the polynomial x⁴ + 1 does not have any real zeroes.
Consider the range of the function x⁴ + 1 for real x.
Find all the zeroes of the polynomial 2x³ - 5x² - 14x + 8, if it is given that one of its zeroes is 4.
Use the given zero to perform polynomial division or factorization to find the remaining zeroes.
Discuss the geometrical meaning of the zeroes of a polynomial with examples.
Visualize the graph of the polynomial and identify where it crosses the x-axis.
If α and β are the zeroes of the polynomial x² - 6x + 8, find the value of α² + β².
Use the identity α² + β² = (α + β)² - 2αβ.
Prove that the polynomial x³ - 4x has exactly three real zeroes and find them.
Factor the polynomial completely to find all its zeroes.
Polynomials - Challenge Worksheet
Push your limits with complex, exam-level long-form questions.
The final worksheet presents challenging long-answer questions that test your depth of understanding and exam-readiness for Polynomials in Class X.
Questions
Given a quadratic polynomial p(x) = 2x² - 8x + 6, find its zeroes and verify the relationship between the zeroes and the coefficients.
Recall the relationship between the coefficients and the sum and product of the zeroes of a quadratic polynomial.
If α and β are the zeroes of the polynomial x² - 5x + 6, find the value of α² + β².
Use the identity α² + β² = (α + β)² - 2αβ.
Prove that the polynomial x³ - 3x² + 3x - 1 has a triple root at x = 1.
Look for a pattern or use the factor theorem to factorize the polynomial.
Find a quadratic polynomial whose zeroes are reciprocal of the zeroes of the polynomial 2x² - 3x - 5.
Use the sum and product of the zeroes to construct the new polynomial.
If one zero of the polynomial (k² + 4)x² + 13x + 4k is reciprocal of the other, find the value of k.
Use the condition that the product of the zeroes is equal to the constant term divided by the coefficient of x².
Show that the polynomial x⁴ + 4x² + 5 has no real zeroes.
Perform a substitution to simplify the polynomial and check the discriminant.
Find the condition that the zeroes of the polynomial p(x) = ax² + bx + c are in the ratio m:n.
Express the sum and product of the zeroes in terms of m, n, and α.
If the polynomial x³ - 3x² + x + 1 has zeroes α, β, γ, find the value of (1/α + 1/β + 1/γ).
Use the sum, sum of products, and product of the zeroes to find the required expression.
Find the cubic polynomial whose zeroes are the squares of the zeroes of the polynomial x³ - 2x² + x + 5.
First find the sum, sum of products, and product of the original zeroes, then find the same for their squares.
Prove that there is no polynomial p(x) with integer coefficients that has p(1) = 2 and p(2) = 3.
Consider the polynomial p(x) - x - 1 and apply the factor theorem.
Real Numbers encompass all rational and irrational numbers, forming a complete and continuous number line essential for various mathematical concepts.
Explore the methods to solve a pair of linear equations in two variables, including graphical, substitution, elimination, and cross-multiplication techniques.
Explore the world of quadratic equations, learning to solve them using various methods like factoring, completing the square, and the quadratic formula.
A chapter that explores sequences where each term after the first is obtained by adding a constant difference, focusing on their properties, nth term, and sum formulas.
