How To Find Sum And Product Of Roots Of A Polynomial With Formula And Examples
There are two roots of a polynomial (quadratic). If α and β are the zeros (0’s) of the quadratic polynomial f(x): ax2+bx+c, then the sum of the roots of the polynomial will be α+β =−b/a. Similarly, it refers to (-coefficient of x)/ (coefficient of x2).
Further, talking about the product of roots, If α and β are the zeros (0’s) of the quadratic polynomial f(x): ax2+bx+c, then the product of the roots of the polynomial will be: αβ= c/a. Similarly, it refers to (constant term)/ (coefficient of x2). This is how we find the sum and product of roots.
Cubic Polynomial Formula
A cubic expression is an algebraic equation of 3rd degree.
A standard form of a cubic function is: f (x) = ax3 + bx2 + cx1 + d. And the cubic equation contains the form of ax3 + bx2 + cx + d = 0, in which a, b and c are the coefficients and d is the constant.
Biquadratic Polynomial
In algebra, a quartic polynomial is a polynomial that consists of a degree of 4 and is in the form ax4+bx3+cx2+dx+e, in which a≠0. While, a biquadratic polynomial is a kind of a quartic polynomial that only contains terms of powers 4, 2, and 0. It is hence of the form:
ax4+bx2+d
What is the Sum of Two Polynomials Known as?
Adding and subtracting polynomials is only about adding and subtracting their like terms. The sum of two monomials is known as a binomial while the sum of three monomials is known as a trinomial. The sum of a finite number of monomials in x is known as a polynomial in x.
Degree of a Polynomial
Algebraic Expressions: An expression, which includes literal numbers, along with the signs of fundamental operations is called an algebraic expression. They may be monomials, having only one term as +6x, -3y, etc. They could also be binomials, i.e. having two terms as p2 +2r, -x2, -2x, etc. They may even be polynomials, i.e. having more than two terms.
For Example: 5x4 +9x3+ 3 is a polynomial. Here 5x4, 9x3, 3 are the terms where 5x4 is a leading term and 3 is a constant term. The coefficients of the polynomial are 5 and 9.
The degree of the polynomial 3x4 + 2x3+ 2 is 3.
Degree of a Zero Polynomial
A zero polynomial can be defined as the one, whose constant function is always equal to zero. So, the degree of the zero polynomial is either indeterminate, or it can be -1.
Degree of a Constant Polynomial
A polynomial can be called the constant which has an unchanged value. There are no variables present in the constant polynomial. One of the examples of a constant polynomial is A(x) = b. A constant polynomial does not have any component and thus the power is always zero here. Hence, the power of the constant polynomial is Zero (0).
Solved Examples on How to Find a Quadratic Polynomial
A Polynomial or a quadratic polynomial is written with variables and powered with coefficients and corresponding powers. Let’s understand how to find the degree of a polynomial.
Example1:
2x2y3z5 + 5x2 + 6xy + 7 y2x5
Solution:
Let’s solve this –
2x2y3z5 – 2+3+5 = 10
5x2 = 2
6xy = 1+1 = 2
7 y2x5 = 2+5 = 7
Thus, the highest degree is 10.
Example2:
Find the degree, constant, and foremost coefficient of the polynomial expression 4x3+ 2x+3.
Solution:
Here, the degree of the polynomial in the expression is 3. Constant is 3. Leading Coefficient is 4. Because the leading variable of the polynomial is 4x3.
Quiz Time
Q1: What is the Degree of This Equation - 5x5+4x+ 3?
The degree of the polynomial is 5
Q2: What is the Degree of This Equation - 12x3 -5x2 + 2?
The degree of this equation is 3
Q3: Identify the Type of Polynomial: 12y3, 5x2
12y3 - Cubic (degree is 3)
5x2 - Quadratic (degree is 2)
Q4: What Is the Degree of This Equation - 4x +12?
The degree of the polynomial is 1
Q5: What is the Degree of This Polynomial 6?
The degree of the polynomial in the expression is 0
Q6: What is the Degree of 1+3z?
The degree of the above polynomial 1 + 3z is 1
Did You Know?
The quadratic formula to determine the roots of a quadratic equation p(x): ax2 + bx + cx is −b±√b2−4ac/2a.
The product of the zeroes is equivalent to the constant term by the coefficient of x2.
The sum of the zeroes of a quadratic polynomial is equivalent to the negative of the coefficient of x by the coefficient of x2.
A polynomial consisting of the value 0 is known as a zero polynomial.
FAQs on Polynomials Sums And Products Of Roots Explained
1. What is the sum and product of roots of a polynomial?
The sum and product of roots of a polynomial are expressions that relate the roots directly to its coefficients. For a quadratic polynomial ax² + bx + c = 0 (a ≠ 0):
- Sum of roots (α + β) = −b/a
- Product of roots (αβ) = c/a
2. What is the formula for the sum and product of roots of a quadratic equation?
The formula for a quadratic equation ax² + bx + c = 0 is α + β = −b/a and αβ = c/a. Here:
- a = coefficient of x²
- b = coefficient of x
- c = constant term
3. How do you find the sum and product of roots without solving the quadratic equation?
You can find the sum and product of roots directly using the coefficients of the quadratic equation. For ax² + bx + c = 0:
- Sum = −b/a
- Product = c/a
- Sum = −7/2
- Product = 3/2
4. What are Vieta’s formulas for polynomials?
Vieta’s formulas relate the sum and product of roots of a polynomial to its coefficients. For a quadratic ax² + bx + c = 0:
- α + β = −b/a
- αβ = c/a
- α + β + γ = −b/a
- αβ + βγ + γα = c/a
- αβγ = −d/a
5. How do you form a quadratic equation when the sum and product of roots are given?
If the sum of roots is S and the product is P, the quadratic equation is x² − Sx + P = 0. Steps:
- Use given sum S
- Use given product P
- Substitute into x² − (sum)x + (product) = 0
6. What is the sum and product of roots of a cubic polynomial?
For a cubic polynomial ax³ + bx² + cx + d = 0, the relationships are given by Vieta’s formulas. If roots are α, β, γ:
- α + β + γ = −b/a
- αβ + βγ + γα = c/a
- αβγ = −d/a
7. Can you give an example of finding the sum and product of roots?
Yes, for the quadratic equation x² − 4x + 3 = 0:
- Sum of roots = −(−4)/1 = 4
- Product of roots = 3/1 = 3
8. Why is the sum of roots equal to −b/a in a quadratic equation?
The sum of roots equals −b/a because of how the polynomial expands from its factored form. If roots are α and β, then:
- a(x − α)(x − β) = a[x² − (α + β)x + αβ]
- −a(α + β) = b
- So, α + β = −b/a
9. How are the sum and product of roots related to the coefficients of a polynomial?
The sum and product of roots are directly determined by the coefficients of the polynomial through Vieta’s formulas. For example, in a quadratic:
- Coefficient of x controls the sum of roots
- Constant term controls the product of roots
10. What are common mistakes when using sum and product of roots formulas?
Common mistakes include incorrect signs and forgetting to divide by the leading coefficient. Key points to remember:
- Always use −b/a for the sum, not just −b
- Always use c/a for the product
- Ensure the equation is in standard form ax² + bx + c = 0
