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Chapter Analysis
Intermediate18 pages • EnglishQuick Summary
This chapter on Polynomials introduces the basic concepts of polynomials in one variable, covering definitions and classifications such as monomials, binomials, and trinomials. It delves into polynomial degrees, the significance of zeroes, and specific theorems like the Remainder and Factor Theorems. The chapter also explores various algebraic identities useful in the factorization of polynomials, providing practice with operations such as addition, subtraction, multiplication, and division of polynomials.
Key Topics
- •Polynomial Basics
- •Monomials, Binomials, Trinomials
- •Polynomial Degrees
- •Zeroes of Polynomials
- •Remainder Theorem
- •Factor Theorem
- •Algebraic Identities
- •Polynomial Operations
Learning Objectives
- ✓Understand the definition and structure of polynomials
- ✓Identify the degree of given polynomial expressions
- ✓Apply the Remainder and Factor Theorems
- ✓Use algebraic identities in polynomial operations
- ✓Perform factorization of polynomial expressions
- ✓Solve equations involving polynomials
Questions in Chapter
Find the value of the polynomial 5x - 4x^2 + 3 at (i) x = 0 (ii) x = -1 (iii) x = 2
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Find p(0), p(1) and p(2) for each of the following polynomials: (i) p(y) = y^2 - y + 1 (ii) p(t) = 2 + t + 2t^2 - t^3 (iii) p(x) = x^3 (iv) p(x) = (x - 1)(x + 1)
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Verify whether the following are zeroes of the polynomial, indicated against them: (i) p(x) = 3x + 1, x = -1/3 (ii) p(x) = 5x - π, x = 4/5 (iii) p(x) = x^2 - 1, x = 1, -1 (iv) p(x) = (x + 1)(x - 2), x = -1, 2...
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Additional Practice Questions
Prove that if x + y + z = 0, then x^3 + y^3 + z^3 = 3xyz
hardAnswer: Expanding both sides using identities and substitutions show that if x + y + z = 0, then x^3 + y^3 + z^3 = 3xyz by (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) = 0.
Factorize the polynomial x^3 - 23x^2 + 142x - 120
mediumAnswer: By synthetic division or trial, find a root such as x = 1, and factor as (x-1)(x^2 - 22x + 120). Then factor the quadratic by methods like completing the square or factor pairing.
Explain the significance of the Remainder Theorem.
mediumAnswer: The Remainder Theorem states that the remainder of the division of the polynomial p(x) by (x-a) is p(a). It helps to find the roots and factors of polynomials easily.
Solve for x: 2x^3 - 3x^2 - 11x + 6 = 0
hardAnswer: Use trial and error to find rational roots or apply synthetic division. Upon finding one root, use it to factor the polynomial further for remaining roots.
Using the identity (x + y)^3 = x^3 + y^3 + 3xy(x + y), evaluate (4 + 2)^3.
easyAnswer: (4 + 2)^3 = 4^3 + 2^3 + 3*4*2*(4+2) = 64 + 8 + 144 = 216.