Introduction
The detailed discussion about zeros of polynomials and how to represent a polynomial on a graph can be found in the chapter Zeros of Polynomial and its Geometrical Meaning. Also, we learnt how to find the value of zeros of 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 cubic polynomial along with zeros and coefficients of a biquadratic polynomial. The zeros and coefficients of a polynomial is related to each other in finding sum and product of roots of the polynomial.
Sum and product of zeros 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 = –
coefficient of x
coefficient of x2
α + β = –
b
a
Product of zeros =
constant term
coefficient of x2
α β =
c
a
Consider a polynomial 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 = –
coefficient of x
coefficient of x2
α + β = –
b
a
(-2) + (-3) = –
5
1
-5 = -5
Product of zeros =
constant term
coefficient of x2
α β =
c
a
(-2)(-3) =
6
1
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 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 = –
coefficient of x2
coefficient of x3
α + β + γ = –
b
a
Sum of product of zeros taken two at a time =
coefficient of x
coefficient of x3
α β + β γ + α γ =
c
a
Product of zeros = –
constant term
coefficient of x3
α β γ = –
d
a
Consider a cubic polynomial 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 = –
coefficient of x2
coefficient of x3
α + β + γ = –
b
a
(-1) + (2) + (-3) = –
2
1
-2 = -2
Sum of product of zeros taken two at a time =
coefficient of x
coefficient of x3
α β + β γ + α γ =
c
a
(-1)(2) + (2)(-3) + (-3)(-1) =
(-5)
1
-2 -6 + 3 = -5
-5 = -5
Product of zeros = –
constant term
coefficient of x3
α β γ = –
d
a
(-1)(2)(-3) = –
(-6)
1
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 = –
coefficient of x3
coefficient of x4
α + β + γ + δ = –
b
a
Sum of product of zeros taken two at a time =
coefficient of x2
coefficient of x4
α β + β γ + δ γ + α δ + δ β + γα =
c
a
Sum of product of zeros taken three at a time = –
coefficient of x
coefficient of x4
αβγ + βγδ + αβδ + αγδ = –
d
a
Product of zeros =
constant term
coefficient of x4
α β γ δ =
e
a
Let x4 – 6x3 – 4x2 + 54x – 45 be a biquadratic polynomial
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 = –
coefficient of x3
coefficient of x4
α + β + γ + δ = –
b
a
(1) + (3) + (5) + (-3) = –
(-6)
1
6 = 6
Sum of product of zeros taken two at a time =
coefficient of x2
coefficient of x4
α β + β γ + δ γ + α δ + δ β + γα =
c
a
(1)(3) + (3)(5) + (5)(-3) + (-3)(1) + (-3)(3) + (5)(1) =
-4
1
3 + 15 – 15 – 3 – 9 + 5 = -4
-4 = -4
Sum of product of zeros taken three at a time
coefficient of x
coefficient of x4
αβγ + βγδ + αβδ + αγδ = –
d
a
(1)(3)(5)+(3)(5)(-3)+(1)(3)(-3)+(1)(5)(-3) = –
54
1
-54 = -54
Product of zeros =
constant term
coefficient of x4
α β γ δ =
e
a
(1)(3)(5)(-3) =
-45
1
-45 = -45
