Explanation And Solution For "Which Is A Solution To X^3 + X^9 = 27"

solve the following pair of linear equations by the substitution method from brainly.in

The Problem Statement

The problem statement is asking us to find the solution to the equation x^3 + x^9 = 27. This is a polynomial equation with degree 9, meaning it has 9 possible solutions. Our goal is to find the specific value of x that satisfies this equation.

Understanding Polynomials

Before we dive into solving this equation, it’s important to understand what a polynomial is. A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, and multiplication. In our case, our polynomial is x^3 + x^9.

Solving the Equation

To solve this equation, we need to find the value of x that makes the equation true. One approach to solving this equation is to use trial and error. We can start by plugging in different values of x and checking if they satisfy the equation. However, this approach can be time-consuming and may not always lead to an exact solution. Another approach is to use algebraic methods. One way to do this is to factor the polynomial. Factoring is the process of finding the expressions that when multiplied together, give us the original expression. In our case, we can factor x^3 + x^9 as x^3(1+x^6). Now we have a product of two factors, and we know that the product of two numbers is equal to 27. Thus, we can set each factor equal to a possible value of the product of the factors which is 27.

Solving for x^3 = 27

If we set x^3 = 27, we get x = 3. This is one possible solution to the equation. However, we still need to check if x^9 also equals 27.

Solving for 1 + x^6 = 27

If we set 1 + x^6 = 27, we get x^6 = 26. Now we can take the sixth root of both sides to get x = (26)^(1/6). This is another possible solution to the equation.

Conclusion

In summary, the equation x^3 + x^9 = 27 has two possible solutions: x = 3 and x = (26)^(1/6). We can verify these solutions by plugging them back into the original equation and checking if they satisfy it. Additionally, we can use algebraic methods such as factoring to simplify the equation and make it easier to solve. Understanding polynomials and their properties is important for solving these types of equations.