Polynomials Class 10 Notes Maths Chapter 2

CBSE Class 10 Maths Notes Chapter 2 Polynomials Pdf free download is part of Class 10 Maths Notes for Quick Revision. Here we have given NCERT Class 10 Maths Notes Chapter 2 Polynomials. According to new CBSE Exam Pattern, MCQ Questions for Class 10 Maths Carries 20 Marks.

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
\(sum\quad of\quad zeros,\alpha +\beta =\frac { -b }{ a } =\frac { -coefficient\quad of\quad x }{ coefficient\quad of\quad { x }^{ 2 } } \)
\(product\quad of\quad zeros,\alpha \beta =\frac { c }{ a } =\frac { constant\quad term }{ coefficient\quad of\quad { x }^{ 2 } } \)

If α, β, γ are the zeroes of the cubic polynomial ax³ + bx² + cx + d = 0, then
\(\alpha +\beta +\gamma =\frac { -b }{ a } =\frac { -coefficient\quad of\quad { x }^{ 2 } }{ coefficient\quad of\quad { x }^{ 3 } } \)
\(\alpha \beta +\beta \gamma +\gamma \alpha =\frac { c }{ a } =\frac { coefficient\quad of\quad { x } }{ coefficient\quad of\quad { x }^{ 3 } } \)
\(\alpha \beta \gamma =\frac { -d }{ a } =\frac { -constant\quad term }{ coefficient\quad of\quad { x }^{ 3 } } \)

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

 

Degree of a Polynomial:

The highest power of the variable in a polynomial is referred to as its degree. For instance, the degree of the polynomial x2+2x+3 is 2 because the highest power of x in the expression is x2. As another example, the degree of the polynomial x8 + 2×6 – 3x + 9 is 8 since the greatest power in the given expression is 8.

Types Of Polynomials

Polynomials can be classified based on the following.
a) Number of terms
b) Degree of the polynomial.

Types of Polynomials Based on the Number of Terms

a) Monomial – A polynomial with just one term. Example: 2x, 6x², 9xy

b) Binomial – A polynomial with two unlike terms. Example: 4x²+x, 5x+4

a) Trinomial – A polynomial with three unlike terms. Example: x²+3x+4

 

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).