How to find roots of polynomial equation?
Understand the Problem
The question is asking how to determine the roots of a polynomial equation, which involves identifying the values of the variable that make the equation equal to zero. The process typically includes methods such as factoring, using the quadratic formula for second-degree polynomials, or applying numerical methods for higher-degree polynomials.
Answer
The roots are $x = 2$ and $x = 3$.
Answer for screen readers
The roots of the polynomial equation are $x = 2$ and $x = 3$.
Steps to Solve
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Identify the polynomial equation Determine the polynomial for which you need to find the roots. For example, let's say the equation is $x^2 - 5x + 6 = 0$.
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Factor the polynomial (if possible) Look for two numbers that multiply to the constant term (6) and add to the coefficient of the linear term (-5). In this case, those numbers are -2 and -3.
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Write the factored form Once you find the numbers, write the polynomial in its factored form: $$ (x - 2)(x - 3) = 0 $$
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Set each factor to zero Now, set each factor equal to zero to find the roots: $$ x - 2 = 0 $$ $$ x - 3 = 0 $$
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Solve for x Now, solve each equation: $$ x = 2 $$ $$ x = 3 $$
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List the roots The roots of the polynomial equation are $x = 2$ and $x = 3$.
The roots of the polynomial equation are $x = 2$ and $x = 3$.
More Information
Finding the roots of a polynomial is essential in various fields, such as physics and engineering, as it often helps in solving real-world problems. By using methods like factoring, you can simplify complex equations to readily find solutions.
Tips
- Forgetting to check if the polynomial can be factored, leading to missed potential roots.
- Ignoring the need to set each factor equal to zero, which results in not finding all roots.
- Confusing the signs when factorizing, which can lead to wrong equations.
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