Maths formulas for class 10 chapter- Polynomials Formula

Math Formulas

A polynomial is an algebraic expression in which the variables have non-integral exponents only.

Example- 3x2­­+ 4y + 2, 5x3+ 3x2 + 4x +2

Degree of a Polynomial

For a polynomial in one variable - The highest exponent on the variable in a polynomial is the degree of the polynomial.

Example: The degree of the polynomial x2+3x+4 is 3, as the highest power of x in the given expression is x2.

Types of Polynomials

  • Number of terms
  • Degree of the polynomial.

Types of Polynomials are based on the Number of terms:

  • Monomial Polynomial– A polynomial with one term. Example: 3x, 8x2, 11xy
  • Binomial – A polynomial that has two terms. Example: 6x2+x, 3x+4
  • Trinomial – A polynomial that has three terms. Example: x2+4x+4

Types of Polynomials are based on the Degree of the Polynomial:

  1. Linear Polynomial- It is a polynomial of degree 1. It is of the form ax+ b where a & b are real numbers and a is not equal to 0.
  2. Quadratic Polynomial- It is a polynomial of degree 2. It is one of the form ax2+ bx + c where a, b & c are real numbers and a is not equal to 0.
  3. Cubic Polynomial- It is a polynomial of degree 3. It is one of the form ax3+ bx2 + cx+ d where a, b & c are real numbers and a is not equal to 0.

Zeroes or roots of a polynomial

It is that value of a variable at which polynomial P(x) becomes zero.

Ex. if polynomial P(x) = x3-6x2+11x-6, Putting x = 1 one get P(1) = 0 then 1 is a zero of polynomial P(x).

  • 0 may be a zero of a polynomial
  • Linear polynomial can have at most one zero
  • Quadratic polynomial can have at most two zeroes
  • Cubic polynomial can have at most three zeroes

Geometrical Meaning of the Zeroes of a Polynomial

  1. The zeroes of a polynomial p(x) are the x-coordinates of the points, where the graph of y = p(x) intersects the x-axis.
  2. Number of zeroes of a polynomial is the number of times the graph intersects the x-axis.

​Relationship between Zeroes and Coefficients of a Polynomial

polynomial

Division Algorithm

The division algorithm states that for any given polynomial p(x) and any non-zero polynomial g(x) there are polynomial q(x) are r(x) such that

p(x) = g(x) × q(x) + r(x)

Dividend = Divisor × Quotient + Remainder

Where r(x) = 0 or degree r(x) < degree g(x)

Number of Zeroes

A polynomial of degree n has at most n zeros.

  • A linear polynomial has one zero,
  • A quadratic polynomial has two zeros.
  • A cubic polynomial has three zeros.

Factorisation of Polynomials

Quadratic polynomials can be factorized by splitting the middle term.

For example, consider the polynomial 2x2−5x+3

Splitting the middle term

The middle term in the polynomial is -5x

Sum = -5

Product = 6

Now, -5 can be expressed as (-2) + (-3) and -2 x -3 = 6

Putting the above value in the gven expression

2x2−5x+3 = 2x2−2x−3x+3

Identify the common factor

2x2−2x−3x+3 = 2x(x−1)−3(x−1)

Taking (x−1) as the common factor, this can be expressed as:

2x(x−1)−3(x−1)=(x−1)(2x−3)

Identities of Algebra

  1. (a+b)2 = a2+2ab+b2
  2. (a−b)2=a2−2ab+b2
  3. (x+a)(x+b) = x2+(a+b)x+ab
  4. a2−b2 = (a+b)(a−b)
  5. a3−b3 = (a−b)(a2+ab+b2)
  6. a3+b3 = (a+b)(a2−ab+b2)
  7. (a+b)3 = a3+3a2b+3ab2+b3
  8. (a−b)3 = a3−3a2b+3ab2−b3

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Maths formulas for class 10 chapter- Polynomials Formula

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