CBSE Class 10 - Maths - CH2 - Polynomials - Study Points

Polynomials

Study Points

1. An expression of the form a₀ + a₁x + a₂x² + ----- + a_nxⁿ where a_n is called a polynomial in variable x of degree n. where; a₀ ,a₁, ----- a_n are real numbers and are called terms/co-efficients of the polynomial and each power of x is a non negative integer.

2. Polynomials in the variable x are denoted by f(x), g(x), h(x) etc.
    e.g. f(x) = a₀ + a₁x + a₂x² + ----- + a_nxⁿ

3. A polynomial p(x) = a (where a is constant) is of degree 0 and is called a constant polynomial.

4. A polynomial p(x) = ax + b is of degree 1 and is called a linear polynomial. e.g. 4x - 3, 5x...

5. A polynomial p(x) = ax² + bx + c of degree 2 and is called a quadratic polynomial.
    e.g. 3x² -x + 5, 1- x²...

6. A polynomial p(x) = ax³ + bx² + cx + d of degree 3 and is called a cubic polynomial.
    e.g. √5x³ - 2x² + 5x -1...



7. Expressions like  5x²  +  1/x ,  x^-2 - x  + 1 are not polynomials

8. Zeroes of a polynomial f(x): A real number α is a zero of the polynomial f(x)
    if and only if f(α) = 0. The graph of y = f(α) intersects the X-axis.

9. A linear polynomial has only one zero (at the most).

10. A Quadratic polynonial at the most has two zeroes.

11. A cubic polynomial  at the most has three zeroes.

12. A polynomial may not have real zeroes. f(x) ≠ 0 for any value of x. Hence f(x) has no real zeroes.

13. A polynomial of degree n can have at most n real zeroes.

14. For any quadratic polynomial ax² + bx + c = 0, a ≠ 0, the graph of this equation i.e. y = ax² + bx + c has one of the two shapes called Parabolas.

15. The parabola for the given quadratic equation will open upwards like ∪ or downwards like ∩ , depending on whether a > 0 or a < 0.

16. If α and β are the zeroes of a quadratic polynomial ax² + bx + c, then

17. A quadratic polynomial whose zeroes are given by α and β is given by:
      p(x) = x² - (α + β)x + αβ
             = x² - (sum of zeroes)x + product of zeroes.

18. If α, β and γ are zeroes of the cubic polynomial ax³ + bx² + cx + d then:
            α+β+γ = -b/a
            αβ + βγ + γα = c/a
            αβγ = -d/a

19. The division algorithm states that given any polynomial p(x) and any non-zero polynomial g(x),
      there are 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).