SUMMARY: The chapter on Polynomials introduces students to the concept of polynomials, their types, and operations on them. KEY TOPICS: definition of polynomials, degree of a polynomial, zeroes of a polynomial, factorization of polynomials, algebraic identities, remainder theorem, factor theorem, types of polynomials (linear, quadratic, cubic), addition and subtraction of polynomials, multiplication of polynomials.
According to the Remainder Theorem, what is the remainder when the polynomial f(x) = 2x^3 - 3x^2 + x - 5 is divided by (x - 2)?
A-5
B-3
C1
D3
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Correct answer: Option 3 — 1
Short Answer Questions5 questions
Q63 Marks
Define a polynomial and give an example.
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A polynomial is an algebraic expression that consists of variables raised to non-negative integer powers and coefficients. For example, 3x^2 + 2x + 1 is a polynomial.
Q73 Marks
What is the degree of the polynomial 4x^3 - 5x + 7?
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The degree of a polynomial is the highest power of the variable in the expression. For the polynomial 4x^3 - 5x + 7, the degree is 3.
Q83 Marks
Explain the Remainder Theorem and provide an example.
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The Remainder Theorem states that when a polynomial f(x) is divided by (x - a), the remainder is f(a). For example, if f(x) = x^2 + 2x + 1 and we divide by (x - 1), the remainder is f(1) = 1^2 + 2(1) + 1 = 4.
Q93 Marks
What are the zeroes of the polynomial f(x) = x^2 - 5x + 6?
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The zeroes of the polynomial f(x) = x^2 - 5x + 6 can be found by factoring it as (x - 2)(x - 3). Thus, the zeroes are x = 2 and x = 3.
Q103 Marks
Use the Factor Theorem to determine if x - 2 is a factor of the polynomial f(x) = x^3 - 3x^2 + 4. Show your working.
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To use the Factor Theorem, we evaluate f(2). f(2) = 2^3 - 3(2^2) + 4 = 8 - 12 + 4 = 0. Since f(2) = 0, x - 2 is a factor of the polynomial.
Long Answer Questions5 questions
Q116 Marks
Define a polynomial and explain its degree. Provide examples of different types of polynomials based on their degrees.
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A polynomial is an algebraic expression that consists of variables raised to non-negative integer powers and coefficients. The degree of a polynomial is the highest power of the variable in the expression. For example, the polynomial 3x^4 + 2x^3 - x + 7 is a polynomial of degree 4, as the highest power of x is 4. Other examples include linear polynomials like 2x + 3 (degree 1) and quadratic polynomials like x^2 - 5x + 6 (degree 2).
Q126 Marks
What are the zeroes of a polynomial? Describe how to find the zeroes of a quadratic polynomial using the factorization method.
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The zeroes of a polynomial are the values of the variable that make the polynomial equal to zero. For a quadratic polynomial of the form ax^2 + bx + c, the zeroes can be found by factorizing the polynomial into the form (px + q)(rx + s) = 0. For instance, to find the zeroes of the polynomial x^2 - 5x + 6, we can factor it as (x - 2)(x - 3) = 0, giving us the zeroes x = 2 and x = 3.
Q136 Marks
Explain the Remainder Theorem and provide an example to illustrate how it is applied to find the remainder of a polynomial when divided by a linear polynomial.
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The Remainder Theorem states that if a polynomial f(x) is divided by a linear polynomial of the form (x - a), the remainder of this division is f(a). For example, consider the polynomial f(x) = 2x^3 - 3x^2 + 4 and we want to find the remainder when it is divided by (x - 1). According to the theorem, we evaluate f(1) = 2(1)^3 - 3(1)^2 + 4 = 2 - 3 + 4 = 3. Thus, the remainder is 3.
Q146 Marks
Discuss the Factor Theorem and how it relates to the zeroes of a polynomial. Provide an example to demonstrate its application.
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The Factor Theorem states that a polynomial f(x) has a factor (x - a) if and only if f(a) = 0. This means that if a is a zero of the polynomial, then (x - a) is a factor of f(x). For example, consider the polynomial f(x) = x^2 - 4. To check if (x - 2) is a factor, we evaluate f(2) = 2^2 - 4 = 0. Since f(2) = 0, by the Factor Theorem, (x - 2) is indeed a factor of f(x), and we can factor the polynomial as (x - 2)(x + 2).
Q156 Marks
Perform the addition and multiplication of the polynomials (3x^2 + 2x - 1) and (x^2 - 4x + 5). Show all steps in your calculations.
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To add the polynomials (3x^2 + 2x - 1) and (x^2 - 4x + 5), we combine like terms: (3x^2 + x^2) + (2x - 4x) + (-1 + 5) = 4x^2 - 2x + 4. For multiplication, we use the distributive property: (3x^2 + 2x - 1)(x^2 - 4x + 5) = 3x^2(x^2) + 3x^2(-4x) + 3x^2(5) + 2x(x^2) + 2x(-4x) + 2x(5) - 1(x^2) + 1(4x) - 1(5). This results in 3x^4 - 12x^3 + 15x^2 + 2x^3 - 8x^2 + 10x - x^2 + 4x - 5, which simplifies to 3x^4 - 10x^3 + 6x^2 + 14x - 5.
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): A polynomial of degree 3 can have at most 3 zeroes.
Reason (R): The degree of a polynomial determines the maximum number of zeroes it can have.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q171 Mark
Assertion (A): The expression 2x^2 + 3x - 5 is a quadratic polynomial.
Reason (R): A quadratic polynomial is defined as a polynomial of degree 2.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q181 Mark
Assertion (A): The factor theorem states that if 'p(x)' is a polynomial and 'a' is a zero of 'p(x)', then (x - a) is a factor of 'p(x)'.
Reason (R): The factor theorem is a specific case of the remainder theorem.
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Correct answer: Option 2 —
Both A and R are true, but R is not the correct explanation of A.
Q191 Mark
Assertion (A): The polynomial 4x^3 - 2x + 1 is a linear polynomial.
Reason (R): A linear polynomial is defined as a polynomial of degree 1.
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Correct answer: Option 3 —
A is true, but R is false.
Q201 Mark
Assertion (A): The sum of two polynomials is always a polynomial.
Reason (R): The set of polynomials is closed under addition.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Statement-Based Questions5 questions
Q211 Mark
Statement 1: A polynomial of degree 2 is called a quadratic polynomial.
Statement 2: The zeroes of a polynomial are the values of the variable that make the polynomial equal to 1.
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Correct answer: Option 2 —
Only Statement 1 is true.
Q221 Mark
Statement 1: The factor theorem states that if a polynomial f(x) has a factor (x - a), then f(a) = 0.
Statement 2: The degree of a polynomial is determined by the highest power of its variable.
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Correct answer: Option 1 —
Both statements are true.
Q231 Mark
Statement 1: The polynomial 3x^3 + 2x^2 - x + 5 is a cubic polynomial.
Statement 2: The sum of two polynomials is always a polynomial of the same degree as the polynomial with the higher degree.
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Correct answer: Option 1 —
Both statements are true.
Q241 Mark
Statement 1: The expression x^2 - 4 can be factored as (x - 2)(x + 2).
Statement 2: A linear polynomial can have more than one zero.
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Correct answer: Option 3 —
Only Statement 2 is true.
Q251 Mark
Statement 1: The remainder theorem can be used to find the remainder of a polynomial when divided by (x - a).
Statement 2: All polynomials are continuous functions.
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Correct answer: Option 1 —
Both statements are true.
