SUMMARY: The chapter on Polynomials in Class 10 Mathematics focuses on the study of polynomials, their properties, and the relationship between their zeros and coefficients. KEY TOPICS: polynomials, degree of a polynomial, zeros of a polynomial, relationship between zeros and coefficients, division algorithm for polynomials, quadratic polynomials, factorization of polynomials, remainder theorem, algebraic identities.
What is the degree of the polynomial 5x^3 - 4x^2 + 2x - 7?
A2
B3
C1
D4
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Correct answer: Option 2 — 3
Q21 Mark
Which of the following is a quadratic polynomial?
A4x^2 + 3x + 5
B2x^3 + x
C7x + 2
D5
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Correct answer: Option 1 — 4x^2 + 3x + 5
Q31 Mark
If p(x) = x^2 - 5x + 6, what are the roots of the polynomial?
A1 and 6
B2 and 3
C-2 and -3
D5 and 1
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Correct answer: Option 2 — 2 and 3
Q41 Mark
Which of the following polynomials is not a monomial?
A3x
B5x^2
Cx^2 + 2x
D7
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Correct answer: Option 3 — x^2 + 2x
Q51 Mark
What is the value of k if the polynomial kx^2 + 4x + 4 has a double root?
A-4
B0
C2
D4
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Correct answer: Option 1 — -4
Short Answer Questions5 questions
Q63 Marks
What is a polynomial? Provide 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
Explain the degree of a polynomial and how to determine it.
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The degree of a polynomial is the highest power of the variable in the expression. For instance, in the polynomial 4x^3 + 2x^2 - x + 5, the degree is 3 because the highest exponent is 3.
Q83 Marks
What are the coefficients in a polynomial? Give an example.
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Coefficients in a polynomial are the numerical factors that multiply the variables. In the polynomial 5x^4 - 3x^2 + 7, the coefficients are 5, -3, and 7.
Q93 Marks
How do you add two polynomials? Illustrate with an example.
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To add two polynomials, combine like terms by adding their coefficients. For example, adding (2x^2 + 3x + 4) and (x^2 - 2x + 1) results in (2x^2 + x^2) + (3x - 2x) + (4 + 1) = 3x^2 + x + 5.
Q103 Marks
What is the factorization of the polynomial x^2 - 5x + 6?
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The polynomial x^2 - 5x + 6 can be factored into (x - 2)(x - 3). This is done by finding two numbers that multiply to 6 and add up to -5, which are -2 and -3.
Long Answer Questions5 questions
Q116 Marks
Explain the Remainder Theorem and demonstrate it by finding the remainder when the polynomial f(x) = 2x^3 - 3x^2 + 4x - 5 is divided by x - 2.
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The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor of the form x - c, the remainder of this division is equal to f(c). To find the remainder of f(x) = 2x^3 - 3x^2 + 4x - 5 when divided by x - 2, we substitute x = 2 into the polynomial. Calculating f(2) gives us 2(2^3) - 3(2^2) + 4(2) - 5 = 16 - 12 + 8 - 5 = 7. Therefore, the remainder is 7.
Q126 Marks
Using the Factor Theorem, determine if x + 3 is a factor of the polynomial f(x) = x^3 + 2x^2 - 5x - 6. Justify your answer with calculations.
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The Factor Theorem states that x - c is a factor of the polynomial f(x) if and only if f(c) = 0. To check if x + 3 is a factor, we substitute x = -3 into f(x). Calculating f(-3) gives us (-3)^3 + 2(-3)^2 - 5(-3) - 6 = -27 + 18 + 15 - 6 = 0. Since f(-3) = 0, it confirms that x + 3 is indeed a factor of the polynomial f(x).
Q136 Marks
Find the zeros of the polynomial p(x) = x^2 - 5x + 6 using the quadratic formula. Show all steps in your solution.
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To find the zeros of the polynomial p(x) = x^2 - 5x + 6, we can use the quadratic formula, which is given by x = (-b ± √(b² - 4ac)) / 2a. Here, a = 1, b = -5, and c = 6. First, we calculate the discriminant: b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1. Since the discriminant is positive, we have two real and distinct roots. Now substituting into the formula gives us x = (5 ± √1) / 2 = (5 ± 1) / 2. Therefore, the zeros are x = 3 and x = 2.
Q146 Marks
Prove that the polynomial f(x) = 3x^4 - 8x^3 + 6x^2 - 4 is divisible by x - 2. Use synthetic division to support your proof.
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To prove that f(x) = 3x^4 - 8x^3 + 6x^2 - 4 is divisible by x - 2, we can use synthetic division. We set up synthetic division with 2 (the root of x - 2) and the coefficients of f(x): 3, -8, 6, 0, -4. Performing synthetic division, we bring down the 3, multiply by 2 to get 6, add to -8 to get -2, multiply by 2 to get -4, add to 6 to get 2, multiply by 2 to get 4, and add to -4 to get 0. Since the remainder is 0, it confirms that f(x) is divisible by x - 2.
Q156 Marks
A polynomial p(x) has roots at x = 1 and x = -2. If p(x) is a quadratic polynomial, write its general form and find the polynomial if it passes through the point (0, -2).
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Given that the roots of the polynomial p(x) are x = 1 and x = -2, we can express p(x) in its factored form as p(x) = k(x - 1)(x + 2), where k is a constant. To find the value of k, we use the condition that the polynomial passes through the point (0, -2). Substituting x = 0 into the polynomial gives us p(0) = k(0 - 1)(0 + 2) = -2k. Setting this equal to -2, we have -2k = -2, which gives k = 1. Therefore, the polynomial is p(x) = (x - 1)(x + 2) = x^2 + x - 2.
Assertion–Reason Questions8 questions
Q161 Mark
Assertion (A): A polynomial of degree 3 can have at most 3 real roots.
Reason (R): The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n roots in the complex number system.
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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 polynomial f(x) = x^4 - 5x^2 + 4 has real roots.
Reason (R): The roots of a polynomial can be complex, and not all polynomials have real roots.
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Correct answer: Option 3 —
A is true, but R is false.
Q181 Mark
Assertion (A): The sum of the roots of the polynomial p(x) = 2x^3 - 3x^2 + x - 5 is given by -b/a.
Reason (R): This is a consequence of Vieta's formulas for polynomials.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q191 Mark
Assertion (A): A polynomial can have more than one term with the same degree.
Reason (R): Like terms can be combined in a polynomial, leading to a single term of that degree.
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Correct answer: Option 4 —
A is false, but R is true.
Q201 Mark
Assertion (A): The polynomial x^2 + 4 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.
Q211 Mark
Assertion (A): The polynomial p(x) = x^3 - 6x^2 + 11x - 6 can be factored into linear factors.
Reason (R): Every cubic polynomial can be expressed as a product of linear factors over the real numbers.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q221 Mark
Assertion (A): The polynomial x^2 - 4 has roots at x = 2 and x = -2.
Reason (R): The roots of a polynomial are the values of x for which the polynomial equals zero.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q231 Mark
Assertion (A): A polynomial of degree n can have at most n-1 turning points.
Reason (R): Turning points occur where the derivative of the polynomial changes sign, which can be at most n-1 times.
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Correct answer: Option 2 —
Both A and R are true, but R is not the correct explanation of A.
Statement-Based Questions5 questions
Q241 Mark
Statement 1: The polynomial x^2 - 4 is a quadratic polynomial.
Statement 2: The polynomial 3x^3 + 2x - 1 is a linear polynomial.
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Correct answer: Option 1 —
Both statements are true.
Q251 Mark
Statement 1: The degree of the polynomial 5x^4 - 3x^2 + 7 is 4.
Statement 2: The polynomial 2x^2 + 3x + 5 has a degree of 3.
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Correct answer: Option 2 —
Only Statement 1 is true.
Q261 Mark
Statement 1: A polynomial can have negative exponents.
Statement 2: The sum of two polynomials is always a polynomial.
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Correct answer: Option 3 —
Only Statement 2 is true.
Q271 Mark
Statement 1: The polynomial 4x^2 + 2x + 1 can be factored into linear factors.
Statement 2: The polynomial x^3 - 2x^2 + x - 2 has at least one real root.
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Correct answer: Option 4 —
Both statements are false.
Q281 Mark
Statement 1: The polynomial 7x^5 is a monomial.
Statement 2: The polynomial x^2 + 1 is a binomial.
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Correct answer: Option 1 —
Both statements are true.
