The detailed discussion about zeros of polynomials and how to represent a polynomial on a graph can be found in the chapter Plotting Of Zeros Of Linear, Quadratic And Cubic Polynomials. Also, we learnt how to find the value of zeros of a polynomial graphically for a linear polynomial, a cubic polynomial and a biquadratic polynomial.
This chapter is about the relationships between zeros and coefficients of a quadratic polynomial, zeros and coefficients of a cubic polynomial along with zeros and coefficients of a biquadratic polynomial. The zeros and coefficients of a polynomial are related to each other in finding the sum and product of roots of the polynomial.
Sum and product of zeros of quadratic
Consider a quadratic polynomial ax2 + bx + c. Let α and β are two zeros of the polynomial. Then, we can find the sum of zeros and product of zeros from coefficients of x and x2 with the following formulas for sum of zeros and product of zeros.
Sum of zeros of quadratic =
Product of zeros of quadratic =
Find sum and product of zeros of x2 + 5x + 6
Zeros of x2 + 5x + 6 are -2 and -3
Here, coefficient of x2 = 1
coefficient of x = 5
constant term = 6
Therefore, a = 1, b = 5, c = 6
Here, α = -2 and β = -3
Sum of zeros of quadratic =
-5 = -5
Product of zeros of quadratic =
6 = 6
Sum and product of zeros of cubic
Now, consider a cubic polynomial p(x)=ax3 + bx2 + cx + d. Let α, β and γ are three zeros of the polynomial. Sum of zeros and product of zeros can be found from the coefficients of x2 and x3 with the following formulas for sum of zeros and product of zeros.
Sum of zeros of cubic =
Sum of product of zeros of cubic taken two at a time
Product of zeros of cubic =
Find sum and product of zeros of x3 + 2x2 - 5x - 6
Zeros of x3 + 2x2 - 5x - 6 are -1, 2 and -3
Here, coefficient of x3 = 1
coefficient of x2 = 2
coefficient of x = -5
constant term = -6
Therefore, a = 1, b = 2, c = -5, d = -6
Here, α = -1, β = 2 and γ=-3
Sum of zeros of cubic =
-2 = -2
Sum of product of zeros of cubic taken two at a time =
(-1)(2) + (2)(-3) + (-3)(-1) =
-2 -6 + 3 = -5
-5 = -5
Product of zeros of cubic =
(-1)(2)(-3) =
6 = 6
Sum and product of zeros of biquadratic
Polynomial p(x)=ax4 + bx3 + cx2 + dx + e, where a ≠ = 0 is a biquadratic polynomial. The graph of y = ax4 + bx3 + cx2 + dx + e intersects the x-axis. These coordinates are the only zeros of the biquadratic polynomial.
Consider a polynomial ax4 + bx3 + cx2 + dx + e. Let α, β, γ and δ are four zeros of the polynomial.
Sum of zeros and product of zeros can be found from the coefficients of x3 and x4 with the following formulas for sum of zeros and product of zeros.
Sum of zeros of biquadratic =
α + β + γ + δ =
Sum of product of zeros of biquadratic taken two at a time
α β + β γ + δ γ + α δ + δ β + γα =
Sum of product of zeros of biquadratic taken three at a time =
αβγ + βγδ + αβδ + αγδ =
Product of zeros of biquadratic =
Find sum and product of zeros of x4 - 6x3 - 4x2 + 54x - 45
Zeros of x4 - 6x3 - 4x2 + 54x - 45 are 1, 3, 5 and -3
Here, coefficient of x4 = 1
coefficient of x3 = -6
coefficient of x2 = -4
coefficient of x = 54
constant term = -45
∴ a = 1, b = -6, c = -4, d = 54, e = -45
Let zeros be, α = 1, β = 3, γ=5 and δ=-3
Sum of zeros of biquadratic =
α + β + γ + δ =
(1) + (3) + (5) + (-3) =
6 = 6
Sum of product of zeros of biquadratic taken two at a time
α β + β γ + δ γ + α δ + δ β + γα =
(1)(3) + (3)(5) + (5)(-3) + (-3)(1) + (-3)(3) + (5)(1) =
3 + 15 - 15 - 3 - 9 + 5 = -4
-4 = -4
Sum of product of zeros of biquadratic taken three at a time =
αβγ + βγδ + αβδ + αγδ =
(1)(3)(5)+(3)(5)(-3)+(1)(3)(-3)+(1)(5)(-3)
-54 = -54
Product of zeros of biquadratic =
-45 = -45
