RBSE Class 10 Maths Notes Chapter 2 Polynomials

These comprehensive RBSE Class 10 Maths Notes Chapter 2 Polynomials will give a brief overview of all the concepts.

RBSE Class 10 Maths Chapter 2 Notes Polynomials

Important Points

1. In a polynomial p(x) of any variable x, the highest power of x is called the degree of the polynomial. For example: 5x + 2 is a polynomial of degree 1 in variable 2y² – 3y + 4 is a polynomial of degree 2 in variable y. x³ – 6a² + x + 9 is a polynomial of degree 3 in variable x.

2. A polynomial of degree 1 is called a linear polynomial, a polynomial of degree 2 is called a quadratic polynomial, and a polynomial of degree 3 is called a cubic polynomial.

3. A quadratic polynomial in variable x is of the form ax² + bx + c where a, b, c are real numbers and a≠0

4. Value of a polynomial – If p(x) is any polynomial in x and k is any real number, then the value obtained by replacing x by k in p(x), is called the value of p(x) at x = k and it is devoted by p(k).

5. Zero of a polynomial—A real number k is said to be a zero of a polynomial p(x), if p(k) = 0. In general,
⇒ p(x) =   ax + b
or p(k) =  ak + b
or p(k) =   0
⇒ ak + b = 0 or k    =  -b/a.

6. The zeroes of polynomial are the x-coordinates of the points, where the graph of y = p(x) Intersect the x-axis.

7. In fact, for any quadratic polynomial ax² + bx + c, a * 0, the graph of the corresponding equation y = ax² + bx + c has one of the two shapes either open upwards like ∪ or open downwords like ∩ depending upon whether
a > 0 or a < 0 (These curves are called parabolas.)

8. A quadratic polynomial can have at most 2 zeroes and a cubic polynomial can have at most 3 zeroes.

Note — In general, given a polynomial p(x) of degree n, the graph of y = p(x) intersects the x-axis at atmost n points. Therefore, a polynomial p(x) of degree n has atmost n zeroes.

9. If α and β are the zeroes of a quadratic polynomial ax² + bx + c, then
α + β = – \(\frac{b}{a}=\frac{-(\text { Coefficient of } x)}{\text { Coefficient of } x^{2}}\)

α β = \(\frac{c}{a}=\frac{\text { Constant term }}{\text { Coefficient of } x^{2}}\)

10. If α, β, γ are the zeroes of the cubic polynomial ax’ + bx² + cx + d = 0, then
α + β + γ = \(\frac{-b}{a}=\frac{-\left(\text { Coefficient of } x^{2}\right)}{\text { Coefficient of } x^{3}}\)

αβ + βγ+ γα  = \(\frac{c}{a}=\frac{\text { Coefficient of } x}{\text { Coefficient of } x^{3}}\)

and αβγ = \(\frac{-d}{a}=\frac{-\text { Constant term }}{\text { Coefficient of } x^{3}}\)

11. According to Euclid’s Division Algorithm, for given polynomial p(x) and non-zero polynomial g(x), these exist two polynomials q(x) and r(x) such that.
p(x) = g(x) q(x) + r(x),
where   r(x) = 0 or degree r(x) < degree g(x)