Presentaton on Polynomials....Class 10
- 2. Polynomials
- 3. Degree of a polynomial
- 4. Constant polynomial A polynomial of degree 1 is called a constant polynomial. For example, f(x) = 7 , g(x) = -1 etc are constant polynomial. Linear Polynomial A polynomial of degree 1 is called a linear polynomial. Example , f(x) = 2x2+3 , g(x) = 3-x2 etc are linear polynomials.
- 6. Bi-quadratic polynomial A forth degree polynomial is called a biquadratic polynomial . For example 3 x4 – 2x3+3x2-12x+10 is a bi quadratic polynomial in variable y . Value of the polynomial If f(x)is a polynomial and α is any real number then the real number obtained by replacing x by α in f(x) , is called the value of f(x) at x=α and is denoted by f(x).
- 7. Zero of the polynomial A real number α is a zero of the polynomial f(x) , if (α)=0 Finding a zero of a polynomial f(x) means solving the polynomial equation f(x)=0. Linear polynomial f(x)=ax+b , a≠0 has only one zero α given by α= -b = - Constant term a Coefficient of x
- 8. Graph of a polynomial In algebraic and set theoretic language the graph of a polynomial f(x) is the collection of all points(x,y),where y=f(x). In geometrical or in grapical language the graph of a polynomial f(x) is a smooth free hand curve passing through points (x1,y1) , (x2,y2),(x3,y3) … etc ,where y1, y2,y3 are the values of the polynomial f(x) at x1, x2, x3… respectively. Graph of Linear polynomial A linear polynomial ax+b=0, a≠0, the graph of y=ax+b is a straight line which intersects the x-axis at exactly one point , namely (-b , 0). Therefore, the linear polynomial ax+b=0 has exactly one zero
- 9. Graph of quadratic polynomial Any quadratic polynomial ax2+bx+c=0 , a≠0 the graph of corresponding equation y=ax2+bx+c has one of the two shapes either open downwards like or open upwards like depending on whether a>0 or a<0. These curves are called parabolas. Graph of cubic polynomial The graph of a cubic polynomial cross the x-axis at least once and almost thrice. Like or .
- 10. Relationship between the zero and the co- efficient of polynomial Consider the quadratic polynomial f(x)= 6x2-x-2 . By the method of splitting the middle term , we have f(x)= 6x2-x-2 = 6x2-4x+3x-2 = 2x(3x-2)+1(3x-2) or , f(x)=(3x-2)(2x+1) f(x)=0 (3x-2)(2x+1)=0 (3x-2)=0 or, 2x+1=0 x=2 or, x= -1 3 2 Hence, the zeroes of 6x2-x-2 are α=2 and β= -1 3 2
- 11. Co efficient of x Sum of its zeroes =α+β = 2- 1=1= Co efficient of x2 3 2 6 Constant term Product of its zeroes=αβ= 2 x -1 = -1 =-2 = Coefficient of x2 . 3 2 3 6 Relationship between the zeroes and coefficients of a quadratic polynomial Let α and β be the zeroes of the quadratic polynomial f(x)=a x2+bx+c. By factor theorem (x-α) and (x-β) are the factors of f(x). f(x)=k(x-α)(x-β), where k is a constant ax2+bx+c=k{x2 - (α+ β)x+ αβ} ax2+bx+c=kx2-k (α+ β)x+kαβ
- 12. Comparing the coefficients of x2 , x and constant terms on both sides, we get a=k,b-k(α+ β)and c= kαβ α+ β= -b and αβ = c a a Coefficient of x Constant term α+ β=-Coefficient of x2 And αβ = Coefficient of x2 Hence , Coefficient of x Sum of the zeroes = -b = Coefficient of x2 a constant term Product of zeroes = c =coefficient of x2
- 13. Relationship between zeroes and Coefficient of a cubic polynomial Let α,β,γ be the zeroes of a cubic polynomial f(x)= ax3+ bx2+cx+d , a≠0.Then , by factor theorem , x- α , x- β, and x- γ are factors of f(x).Also , f(x),being a cubic polynomial ,cannot have more than three linear factors. f(x)= k(x- α)(x- β)(x- γ) ax3+ bx2+cx+d = k(x- α)(x- β)(x- γ) ax3+ bx2+cx+d = k{x3 -(α+β+γ) x2+(αβ+ βγ+ γα)x- αβγ} ax3+ bx2+cx+d = kx3-k(α+β+γ) x2+k (αβ+ βγ+ γα)x – kαβγ Comparing the coefficients of x3 , x2 , x and constant terms on both sides , we get a=k ,b= -k(α+β+γ) , c =k (αβ+ βγ+ γα)x and d = -k(αβγ) α+β+ γ = -b/a αβ+ βγ+ γα = c/a αβγ = -d/a
- 14. - b Coefficient of x sum of the zeroes= a = Coefficient of x3 product of the zeroes taken two at a time = c = Coefficient of x a Coefficient of x3 Product of the zeroes = -d = -Constant term a Coefficient of x3
- 15. Division algorithm for polynomial Dividend=quotient x divisor +remainder… This is Euclid’s division lemma. Division algorithm If f(x) and g(x) are ant two polynomials with g(x)≠0, then we can always find polynomials q(x) and r(x) such that f(x)= g(x) x q(x) + r(x) , where r(x)=0 or degree r(x)=0 or degree r(x)< degree g(x). REMARK : If r(x)=0, then polynomial g(x) is a factor of polynomial f(x).