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Chapter Analysis
Intermediate14 pages • EnglishQuick Summary
This chapter on polynomials introduces students to the concept of polynomials, their degrees, and types (linear, quadratic, and cubic). It goes on to explain the geometrical meaning of the zeroes of a polynomial and their relation to the graph of the polynomial. The chapter also discusses the relationship between the coefficients of a polynomial and its zeroes, providing methods to find and verify zeroes through exercises.
Key Topics
- •Types of polynomials: linear, quadratic, and cubic
- •Geometrical meaning of zeroes of a polynomial
- •Relation between coefficients and zeroes of polynomials
- •Methods for finding zeroes of polynomials
- •Graphical representation of polynomials
- •Verification of polynomial identities
Learning Objectives
- ✓Understand the definition and classification of polynomials
- ✓Learn how to graphically interpret the zeroes of polynomials
- ✓Explore the relationship between polynomial coefficients and their zeroes
- ✓Master the methods for finding zeroes of quadratic and cubic polynomials
- ✓Apply the polynomial division algorithm
Questions in Chapter
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (i) x^2 – 2x – 8 (ii) 4s^2 – 4s + 1 (iii) 6x^2 – 3 – 7x (iv) 4u^2 + 8u (v) t^2 – 15 (vi) 3x^2 – x – 4
Answer: Not provided
Page 23
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (i) -1/4, 1 (ii) 1/2, 3 (iii) 0, 5 (iv) 1, 1 (v) -1/4, -1/4 (vi) 4, 1
Answer: Not provided
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Additional Practice Questions
What is a polynomial, and how are polynomials classified?
easyAnswer: A polynomial is an algebraic expression with one or more terms, each consisting of a constant multiplied by a variable raised to a non-negative integer power. Polynomials are classified based on their degree: linear (degree 1), quadratic (degree 2), cubic (degree 3), etc.
Explain the geometrical meaning of the zeroes of a quadratic polynomial.
mediumAnswer: The zeroes of a quadratic polynomial are the x-coordinates where its graph intersects the x-axis. These points correspond to the solutions of the quadratic equation obtained by setting the polynomial equal to zero.
How can the relationship between the zeroes and coefficients of a quadratic polynomial be verified?
mediumAnswer: For a quadratic polynomial ax^2 + bx + c, the sum of zeroes is given by -b/a, and the product of zeroes is c/a. Verification involves substituting these expressions with known zeroes and comparing the results with the given polynomial coefficients.
What is the significance of the degree of a polynomial in relation to its zeroes?
easyAnswer: The degree of a polynomial determines the maximum number of zeroes it can have. A polynomial of degree n has at most n zeroes, which are the x-coordinates where its graph intersects the x-axis.
Define a cubic polynomial and provide an example.
easyAnswer: A cubic polynomial is a polynomial of degree 3, and it is of the form ax^3 + bx^2 + cx + d, where a, b, c, and d are constants with a ≠ 0. An example is p(x) = 2x^3 - 3x^2 + x - 5.
How can you find the zeroes of a cubic polynomial?
hardAnswer: To find the zeroes of a cubic polynomial, one can factorize the polynomial by finding one zero through trial or the Rational Root Theorem and then use synthetic division or factorization to find the remaining zeroes.