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 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.
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
-5 = -5
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 product of zeros of cubic taken two at a time
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
-2 = -2
Sum of product of zeros of cubic taken two at a time
-2 -6 + 3 = -5
-5 = -5
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 =
Sum of product of zeros of biquadratic taken two at a time
Sum of product of zeros of biquadratic taken three at a time =
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 =
6 = 6
Sum of product of zeros of biquadratic taken two at a time
(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 =
(1)(3)(5)+(3)(5)(-3)+(1)(3)(-3)+(1)(5)(-3)
-54 = -54
Product of zeros of biquadratic =
-45 = -45
Frequently Asked Questions
1) What is the relationship between coefficient of a quadratic polynomial and its zero?
Solved Examples
1) Verify its relationship between coefficient of polynomial and its zeros for a polynomial x2 - 9.
Solution
Here p(x) = x2 - 9
compare it with ax2 + bx + c
p(x) = x2 + 0x - 9
Here a = 1, b = 0, c = -9
Sum of zeros =
-3 + 3 = 0
=
= -
Product of zeros =
= (-3)(3) = -9
=
=
2) Form a quadratic polynomial whose sum of zeros is 5 and product of zeros is 6.
Solution
Sum of zeros = 5
Product of zeros = 6
As we know, quadratic polynomial is the form of x2 - (sum of zeros)x + product of
zeros
By putting the above values, it becomes x2 - 5x + 6
Hence, x2 - 5x + 6 is a quadratic polynomial.
3) Verify relationship between coefficient of polynomial and its zeros if x3 - 9x2 - 12x + 20 has zeros -2, 1 and 10.
Solution
Here, compare x3 - 9x2 - 12x + 20 with ax3 + bx2 + cx +
d
a = 1, b = -9, c = -12, d = 20
Zeros are -2, 1 and 10 (given)
α = -2
β = 1
γ = 10
Sum of zeros of cubic =
∴ α + β + γ = -2 + 1 + 10
= 9
=
=
Sum of product of zeros of cubic taken two at a time
αβ + βγ + γα = (-2)(1) + (1)(10) + (10)(-2)
= -2 + 10 -20
= -12
=
=
Product of zeros of cubic =
αβγ = (-2)(1)(10)
= -20
=
=
