Relation between roots and coefficients of different types of polynomial
Roots and Coefficients are the terms from quadratic equation or polynomial. There is also a relation between them. But before we establish the relation between roots and coefficients for different polynomial. We should know the meaning of these terms so that, it will be easy to understand the relation. Derivation of relation between roots and coefficients are also discuss in later part. So lets start with Roots of the quadratic equation.
What is roots of the quadratic equation
Roots of a quadratic equation is defined as, those numbers when put in a quadratic equation in place of variable the value become zero. In other words those number called the roots of the quadratic equation when placed in place of variable in quadratic equation and value comes equal to zero. For example: A real number π is called root of the quadratic equation ax2 + bx = c = 0, a ≠ 0 if ax2 + bx + c = 0.
If π is a root of ax2 + bx + c = 0, then we say that x = π satisfies the equation ax2 + bx + c = 0. please note that the roots of the quadratic equation ax2 + bx + c = 0 are called the zeros of the polynomial
ax2 + bx + c.
Let π and π are the two zeros or roots of the quadratic equation ax2 + bx + c = 0. then,
π = (-b + √b2 - 4ac) / 2a and π = (-b - √b2 - 4ac) / 2a
Please note that b2 - 4ac also called the discriminant. It is denoted by D.
D = b2 - 4ac
As we know that the roots of the quadratic equation called the zeros of the polynomial. So In polynomial roots are use in the form of sum and product of the zeros.
Let a polynomial ax2 + bx + c having π and π be the zeros of the polynomial then,
sum of the zeros (π + π) = (-b + √b2 - 4ac) / 2a + (-b - √b2 - 4ac) / 2a
(π + π) = - b /a
Product of zeros (π.π) = (-b + √b2 - 4ac) / 2a ✕ (-b - √b2 - 4ac) / 2a
(π.π) = c /a
Nature of roots
Case 1:- when D > 0
In this case, the roots are real and distinct.
Case 2:- when D = 0
In this case, roots are real and equal.
Case 3:- when D < 0
In this case, the roots are imaginary, and we say that the given equation has no real roots.
What is coefficients of quadratic equation
Coefficients of quadratic polynomial is the constant number which can never be change. For example let P(x) = ax2 +bx + c = 0 be the polynomial. Here a, b and c are the constant term which is also called the coefficients. please note that a ≠ 0.
In other words An expression of the form ax2 + bx + c = 0, where a ≠ 0, called a polynomial having a, b, and c are coefficients and x2, x are variable.
Relation between roots and coefficients of quadratic equation
The relationship between roots and coefficients is, In a quadratic equation the sum of the roots is equal to the negative of the coefficient of second term divided by the coefficient of first term. while the product of roots is equal to the third term divided by the first term.
Let a polynomial ax2 + bx + c having π and π be the zeros of the polynomial then,
sum of the zeros (π + π) = (-b + √b2 - 4ac) / 2a + (-b - √b2 - 4ac) / 2a
(π + π) = - b /a
Product of zeros (π.π) = (-b + √b2 - 4ac) / 2a ✕ (-b - √b2 - 4ac) / 2a
(π.π) = c /a
Here we consider a polynomial ax2 + bx + c = 0. As we know that this polynomial is quadratic so there is only two zeros possible. Which we consider the name π and π.
Now, If π and π are the zeros of the polynomial then (x - π) and (x -π) be the factor of the polynomial ax2 + bx + c = 0. So, this polynomial must be equal to the product of these two factors.
∴ ax2 + bx + c = (x - π) (x -π)
Multiply any constant term in right side so that polynomial can be compared.
ax2 + bx + c = k (x - π) (x -π)
ax2 + bx + c = k. {x2 - (π + π) x + ππ}
ax2 + bx + c = kx2 - k(π + π) x + k(ππ)
On comparing coefficients of like powers of x on both sides, we get:
k = a, - k(π + π) = b and k(ππ) = c
Now, replace 'k' by 'a' because k = a.
= - a(π + π) = b and a(ππ) = c
= (π + π) = - b/a and ππ = c/a
Therefore,
Sum of zeros = -(coefficient of x) divided by coefficient of x2.
Product of zeros = constant term divided by coefficient of x2.
Relation between roots and coefficients question
Example 1:- Find the zeros of the polynomial f(x) = x2 + 7x + 12 and verify the relation between its zeros and coefficients.
we have,
f(x) = x2 + 7x + 12
= x2 + 4x + 3x + 12
= x(x + 4) + 3(x + 4)
= (x + 4)(x + 3)
∴ f(x) = 0 = (x + 4)(x + 3) = 0
x = -4 and x = -3
Hence, -4 and -3 are the zeros of the polynomial f(x) = x2 + 7x + 12.
Sum of zeros = -4 + (-3) = -7/1
Product of zeros = -4.-3 = 12/1
Example 2:- Find the zeros of the polynomial f(x) = 2x2 + 5x - 12 and verify the relation between its zeros and coefficients.
we have,
f(x) = 2x2 + 5x - 12
= 2x2 + 8x - 3x - 12
= 2x(x + 4) - 3(x + 4)
= (x + 4) (2x - 3)
f(x) = 0 = (x + 4) (2x - 3) = 0
x = -4 and x = 3/2 are the zeros of the polynomial f(x) = 2x2 + 5x - 12.
Sum of zeros = -4 + 3/2 = -5/2
Product of zeros = (-4) (3/2) = -12/2
Relation between roots and coefficients of cubic polynomial
The equation is in the form of ax3 + bx2 + cx + d = 0 called cubic polynomial. If πΌ, Ξ², and π€ are the roots of a cubic equation ax3 + bx2 + cx + d = 0 then,
πΌ + Ξ² + π€ = -b/a
πΌΞ² + Ξ²π€ + π€πΌ = c/a
πΌΞ²π€ = -d/a
Here, alpha, beta, and gamma are the three roots of the given cubic polynomial. there are total three relation between roots and coefficients of cubic polynomial.
First relation is describing as the sum of all roots are equal to the negative of the coefficient of second term divided by coefficient of first term.
The second relation is describing as the sum of the products of (alpha and beta, beta and gamma, gamma and alpha) are equal to the third term divided by first term.
The last third relation is describing as the products of all three roots alpha, beta and gamma is equal to the negative of last term divided by first term.
Relation between roots and coefficients of biquadratic polynomial
If πΌ, Ξ², π€, and Ξ΄ are the roots of the biquadratic equation ax4 + bx3 + cx2 + dx + e = 0. then,
πΌ + Ξ² + π€ + Ξ΄ = -b/a
πΌΞ² + Ξ²π€ + π€Ξ΄ + πΌπ€ + πΌΞ΄ + Ξ²Ξ΄ = c/a
πΌΞ²π€ + πΌπ€Ξ΄ + πΌΞ²Ξ΄ + Ξ²π€Ξ΄ = -d/a
πΌΞ²π€Ξ΄ = e/a
Here, alpha, beta, gamma and delta are the roots of the given biquadratic polynomial. there are total four relation between roots and coefficients of biquadratic equation.
First relation is the sum of all four roots alpha, beta, gamma and delta are equal to the negative of second term divided by first term.
Second relation is the sum of the products of (alpha and beta, beta and gamma, gamma and theta, alpha and gamma, alpha and theta, beta and theta) are equal to the third term divided by first term.
Third relation is the sum of the products of πΌΞ²π€ and πΌπ€Ξ΄ and πΌΞ²Ξ΄ and Ξ²π€Ξ΄ are equal to negative of third term divided by first term.
The last fourth relation is the products of all four roots alpha, beta, gamma and theta are equal to the fourth term divided by first term.