CBSE Class 10 Maths Notes & Important Questions Chapter 2 Polynomials

CBSE Class 10 Maths Notes & Important Questions Chapter 2 Polynomials

CBSE Class 10 Maths Notes Chapter 2 Polynomials

  1. A polynomial is an expression that consists of variables and coefficients and involves only the operations of addition, subtraction, and multiplication.
  2. The degree of a polynomial is the highest power of the variable present in the polynomial.
  3. A polynomial of degree 0 is called a constant polynomial.
  4. A polynomial of degree 1 is called a linear polynomial.
  5. A polynomial of degree 2 is called a quadratic polynomial.
  6. A polynomial of degree 3 is called a cubic polynomial.
  7. The standard form of a quadratic polynomial is ax² + bx + c, where a, b, and c are constants and a ≠ 0.
  8. The factor theorem states that if a polynomial f(x) is divided by (x – a), then the remainder is f(a).
  9. The remainder theorem states that if a polynomial f(x) is divided by (x – a), then the remainder is zero if and only if f(a) = 0.
  10. A polynomial with real coefficients can be factored into linear and quadratic factors with real coefficients.
  11. The zeroes of a polynomial are the values of the variable for which the polynomial becomes zero.
  12. The Fundamental Theorem of Algebra states that every polynomial equation of degree n has n complex roots, including repeated roots.
  13. The sum of the zeroes of a polynomial is equal to the negative coefficient of the term of the second highest degree, divided by the coefficient of the term of the highest degree.
  14. The product of the zeroes of a polynomial is equal to the constant term divided by the coefficient of the term of the highest degree.

CBSE Class 10 Maths Important Questions Chapter 2 Polynomials

  1. State the factor theorem.
  2. State the remainder theorem.
  3. Find the zeroes of the polynomial x^3 – 3x^2 – 4x + 12, if two of its zeroes are 2 and -2.
  4. Factorize the polynomial 6x^2 – 5x – 6.
  5. Find a polynomial of degree 3 with zeroes 1, 2 and -3.
  6. If (x – 2) is a factor of the polynomial 2x^3 + ax^2 + bx – 12, find the value of a, b, and the third factor.
  7. If the zeroes of the polynomial ax^2 + bx + c are equal, find the value of b^2 – 4ac.
  8. Find the value of k for which x – 1 is a factor of the polynomial 3x^3 – 4kx^2 + 3x + 2k.
  9. If the zeroes of the polynomial x^3 – 2x^2 – 5x + 6 are a, b and c, find the value of a + b + c.
  10. Factorize the polynomial x^4 – 4x^3 + 6x^2 – 4x + 1.

CBSE Class 10 Maths Important Questions Answers Chapter 2 Polynomials

  1. The factor theorem states that if a polynomial f(x) is divided by (x – a), then the remainder is f(a).
  2. The remainder theorem states that if a polynomial f(x) is divided by (x – a), then the remainder is zero if and only if f(a) = 0.
  3. Using the given zeroes, the polynomial can be written as (x-2)(x+2)(x-3). Thus, the zeroes of the polynomial are 2, -2 and 3.
  4. The polynomial can be factorized as (2x+3)(3x-2).
  5. A polynomial of degree 3 with zeroes 1, 2 and -3 is (x-1)(x-2)(x+3).
  6. Since (x-2) is a factor of the given polynomial, f(2) = 0. Therefore, we have 16a + 4b – 12 = 0. Also, the third factor is (2x + p), where p is a constant. Using the given polynomial, we have (2x+p)(x-2)^2 = 2x^3 + ax^2 + bx – 12. Expanding this equation, we get p = -6, a = 12, and b = -25.
  7. If the zeroes of the polynomial are equal, then the discriminant b^2 – 4ac is equal to zero. Therefore, we have b^2 – 4ac = 0.
  8. If x – 1 is a factor of the polynomial, then f(1) = 0. Using the given polynomial, we have 3 – 4k + 3 + 2k = 0. Therefore, we have k = 1/2.
  9. Using the given zeroes, the polynomial can be written as (x-a)(x-b)(x-c) = x^3 – (a+b+c)x^2 + (ab+bc+ca)x – abc. Equating the coefficients with the given polynomial, we get a + b + c = 2, ab + bc + ca = -5, and abc = -6. Solving these equations, we get a = 1, b = -2, and c = -3. Therefore, a + b + c = -4.
  10. The given polynomial can be written as (x-1)^4.

CBSE Class 10 Maths Important Questions Answers MCQs Chapter 2 Polynomials

The degree of the polynomial 5x^4 – 3x^2 + 2 is:
a) 4
b) 3
c) 2
d) 1
Answer: a) 4

Which of the following is not a polynomial?
a) x^3 + 2x^2 – x + 5
b) 3x – 2
c) 5x^2 – 7x + 1/x
d) 2x^2 + 5x – 1
Answer: c) 5x^2 – 7x + 1/x

If the zeroes of the polynomial x^3 – 3x^2 + x + 2 are a, b and c, then their sum is:
a) -1
b) 1
c) 2
d) 3
Answer: b) 1

If (x – 2) is a factor of the polynomial x^3 + ax^2 + bx + 6, then the value of b is:
a) -6
b) 6
c) 2
d) -2
Answer: a) -6

The polynomial 3x^2 – 2x + 1 is a:
a) Quadratic polynomial
b) Linear polynomial
c) Cubic polynomial
d) Zero polynomial
Answer: a) Quadratic polynomial

If (x – 1) is a factor of the polynomial 2x^3 + 3x^2 – 5x + k, then the value of k is:
a) 1
b) -1
c) 0
d) 2
Answer: b) -1

The zeroes of the polynomial x^2 – 4x + 4 are:
a) 2, 2
b) 2, -2
c) 4, -4
d) -2, -2
Answer: a) 2, 2

If a polynomial has degree 2, then it can have:
a) 0 zeroes
b) 1 zero
c) 2 zeroes
d) 3 zeroes
Answer: c) 2 zeroes

The zeroes of the polynomial 2x^2 + 5x – 3 are:
a) -1/2, 3/2
b) 1/2, -3/2
c) -3/2, -1/2
d) 3/2, -1/2
Answer: a) -1/2, 3/2

If a polynomial has degree 3, then it can have:
a) 1 zero
b) 2 zeroes
c) 3 zeroes
d) 4 zeroes
Answer: c) 3 zeroes

CBSE Class 10 Maths Notes Chapter 2 Polynomials

  • “Polynomial” comes from the word ‘Poly’ (Meaning Many) and ‘nomial’ (in this case meaning Term)-so it means many terms.
  • A polynomial is made up of terms that are only added, subtracted or multiplied.
  • A quadratic polynomial in x with real coefficients is of the form ax² + bx + c, where a, b, c are real numbers with a ≠ 0.
  • Degree – The highest exponent of the variable in the polynomial is called the degree of polynomial. Example: 3x3 + 4, here degree = 3.
  • Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomial respectively.
  • A polynomial can have terms which have Constants like 3, -20, etc., Variables like x and y and Exponents like 2 in y².
  • These can be combined using addition, subtraction and multiplication but NOT DIVISION.
  • The zeroes of a polynomial p(x) are precisely the x-coordinates of the points, where the graph of y = p(x) intersects the x-axis.

If α and β are the zeroes of the quadratic polynomial ax² + bx + c, then
sumofzeros,α+β=−ba=−coefficientofxcoefficientofx2
productofzeros,αβ=ca=constanttermcoefficientofx2

If α, β, γ are the zeroes of the cubic polynomial ax3 + bx2 + cx + d = 0, then
α+β+γ=−ba=−coefficientofx2coefficientofx3
αβ+βγ+γα=ca=coefficientofxcoefficientofx3
αβγ=−da=−constanttermcoefficientofx3

Zeroes (α, β, γ) follow the rules of algebraic identities, i.e.,
(α + β)² = α² + β² + 2αβ
∴(α² + β²) = (α + β)² – 2αβ

DIVISION ALGORITHM:
If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then
p(x) = g(x) × q(x) + r(x)
Dividend = Divisor x Quotient + Remainder

Remember this!

  • If r (x) = 0, then g (x) is a factor of p (x).
  • If r (x) ≠ 0, then we can subtract r (x) from p (x) and then the new polynomial formed is a factor of g(x) and q(x).

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