Find Equation Given Roots Calculator

Equation from Roots Calculator

Enter the roots of the polynomial and the leading coefficient 'a' to find the equation.

What is a Find Equation Given Roots Calculator?

A find equation given roots calculator is a tool used to determine the polynomial equation when its roots (the values of x for which the polynomial equals zero) are known. If you know the points where a polynomial graph crosses the x-axis, this calculator helps you find the algebraic expression of that polynomial. It's particularly useful for quadratic (degree 2), cubic (degree 3), and higher-degree polynomials.

This calculator is beneficial for students learning algebra, teachers creating examples, and engineers or scientists who might encounter roots of equations in their work. A common misconception is that only one equation corresponds to a given set of roots. However, there are infinitely many equations if the leading coefficient 'a' is not specified; they all belong to the same family, `a * P(x) = 0`, where `P(x)` is the base polynomial with those roots and leading coefficient 1.

Find Equation Given Roots Formula and Mathematical Explanation

The fundamental principle used by the find equation given roots calculator is the Factor Theorem. If `r` is a root of a polynomial `P(x)`, then `(x – r)` is a factor of `P(x)`.

If a polynomial of degree `n` has roots `r1, r2, …, rn`, then the polynomial can be expressed as:

P(x) = a(x - r1)(x - r2)...(x - rn)

where `a` is the leading coefficient (the coefficient of the x^n term).

For a Quadratic Equation (Degree 2):

If the roots are `r1` and `r2`, and the leading coefficient is `a`, the equation is:

a(x - r1)(x - r2) = 0

Expanding this gives:

a(x² - r1*x - r2*x + r1*r2) = 0

a(x² - (r1 + r2)x + r1*r2) = 0

ax² - a(r1 + r2)x + a(r1*r2) = 0

Comparing this to the standard quadratic form `ax² + bx + c = 0`, we have:

• `b = -a(r1 + r2)` (b = -a * sum of roots)
• `c = a(r1 * r2)` (c = a * product of roots)

Our find equation given roots calculator uses these relationships.

For a Cubic Equation (Degree 3):

If the roots are `r1`, `r2`, and `r3`, and the leading coefficient is `a`:

a(x - r1)(x - r2)(x - r3) = 0

Expanding this leads to `ax³ + bx² + cx + d = 0`, where `b`, `c`, and `d` depend on `a` and the sums and products of the roots taken one, two, and three at a time (Vieta's formulas).

Variables Table:

Variable	Meaning	Unit	Typical Range
x	The variable in the polynomial	None	Real or Complex Numbers
r1, r2, …	The roots of the polynomial	None	Real or Complex Numbers
a	Leading coefficient	None	Non-zero Real or Complex Numbers
b, c, d…	Other coefficients of the polynomial	None	Real or Complex Numbers
Sum of roots	r1 + r2 + …	None	Real or Complex Numbers
Product of roots	r1 * r2 * …	None	Real or Complex Numbers

Table 1: Variables involved in finding an equation from its roots.

Practical Examples (Real-World Use Cases)

Example 1: Quadratic Equation

Suppose the roots of a quadratic equation are 2 and -5, and the leading coefficient `a` is 1.

• r1 = 2, r2 = -5, a = 1
• Sum of roots = 2 + (-5) = -3
• Product of roots = 2 * (-5) = -10
• Equation: 1(x² – (-3)x + (-10)) = 0 => x² + 3x – 10 = 0

Using the find equation given roots calculator with inputs r1=2, r2=-5, a=1 gives `x^2 + 3x – 10 = 0`.

Example 2: Cubic Equation

Find the cubic equation with roots 0, 1, and 3, and a leading coefficient `a` of 2.

• r1 = 0, r2 = 1, r3 = 3, a = 2
• Factors: (x – 0), (x – 1), (x – 3)
• Equation: 2 * x * (x – 1)(x – 3) = 0
• 2x(x² – 3x – x + 3) = 0
• 2x(x² – 4x + 3) = 0
• 2x³ – 8x² + 6x = 0

The find equation given roots calculator can handle this as well.

How to Use This Find Equation Given Roots Calculator

1. Select Degree: Choose the degree of the polynomial (2 for quadratic, 3 for cubic, etc.) from the dropdown. This will show the correct number of root input fields.
2. Enter Roots: Input the known roots (r1, r2, …) into the respective fields. These are the values of x where the polynomial equals zero.
3. Enter Leading Coefficient (a): Input the desired leading coefficient. If you want the simplest monic polynomial (leading coefficient is 1), enter 1. A non-zero value is required.
4. Calculate: The calculator automatically updates the equation and intermediate results as you type. You can also click "Calculate Equation".
5. View Results: The primary result is the polynomial equation. You'll also see intermediate values like the sum and product of roots (for quadratics) and the coefficients.
6. See the Graph: For quadratic and cubic equations, a graph is shown, plotting the polynomial and highlighting the roots on the x-axis.
7. Reset: Use the "Reset" button to clear inputs and return to default values.
8. Copy Results: Use the "Copy Results" button to copy the equation and key values.

The find equation given roots calculator provides the equation in the standard form `ax^n + bx^(n-1) + … = 0`.

Key Factors That Affect the Equation

• The Roots Themselves: The values of the roots directly determine the factors `(x-r)`.
• Number of Roots: This determines the degree of the polynomial. Two distinct roots (for a quadratic) will give a different form than three distinct roots (for a cubic).
• Multiplicity of Roots: If a root appears more than once (e.g., roots 2, 2, -1), the factor `(x-2)` appears squared: `a(x-2)²(x+1)=0`. Our calculator assumes distinct roots as entered.
• Real vs. Complex Roots: If roots are complex conjugates (e.g., 2+3i and 2-3i), the resulting polynomial will have real coefficients. Our current calculator is primarily designed for real roots entered as numbers.
• The Leading Coefficient (a): This scales the entire polynomial but does not change the roots. Changing 'a' gives a different equation with the same roots. If `a=1`, the polynomial is monic.
• Order of Roots: The order in which you enter the roots does not affect the final expanded equation.

Understanding these factors helps in interpreting the results from the find equation given roots calculator. For more on quadratic equations, see our quadratic formula calculator.

Frequently Asked Questions (FAQ)

1. Can this calculator find cubic or higher-degree equations?

Yes, our find equation given roots calculator allows you to select the degree (number of roots) up to 4, so it can find quadratic, cubic, and quartic equations given their roots.

2. What if some roots are repeated (multiplicity)?

If a root is repeated, you simply enter it the number of times it appears. For example, for roots 2, 2, -1, you would enter 2 as root 1, 2 as root 2, and -1 as root 3 if finding a cubic.

3. What if the roots are complex numbers?

This calculator is primarily designed for real number inputs for the roots. For complex roots, you would need a calculator that can handle complex number arithmetic during expansion. If complex roots come in conjugate pairs, the final equation with a real 'a' will have real coefficients.

4. Is the equation found unique?

The equation is unique ONLY if the leading coefficient 'a' is specified. If 'a' is not specified, there is an infinite family of equations `k * P(x) = 0` that share the same roots, where `k` is any non-zero constant.

5. What does the leading coefficient 'a' do?

The leading coefficient 'a' scales the polynomial vertically. It stretches or compresses the graph and can reflect it across the x-axis if 'a' is negative, but it doesn't change the x-intercepts (the roots).

6. Can I find the equation if only the sum and product of roots are known (for a quadratic)?

Yes, for a quadratic with `a=1`, the equation is `x² – (sum of roots)x + (product of roots) = 0`. If `a` is different from 1, it's `ax² – a(sum)x + a(product) = 0`. You can explore polynomial relationships further.

7. What if one of the roots is zero?

If one root is zero, then `x` is a factor of the polynomial. For example, if roots are 0 and 3, the equation is `a * x * (x-3) = 0` or `ax² – 3ax = 0`.

8. How is this related to Vieta's formulas?

Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. This calculator uses the reverse idea: given the roots, it finds the coefficients by expanding `a(x-r1)(x-r2)…`. For more on polynomial structures, check our resources.

Related Tools and Internal Resources

Explore these other calculators and resources:

• {related_keywords[0]} Calculator: If you have the equation and want to find the roots.
• Polynomial Long Division Calculator: To divide polynomials.
• {related_keywords[2]} Solver: Tools to find roots based on coefficients.
• Graphing Calculator: To visualize polynomial functions and their roots.
• {related_keywords[4]} Explained: Understanding the relationship between roots and factors.
• Completing the Square Calculator: Another method related to quadratic equations.