Question

Find the real solutions of each equation. $$ x^{6}+7 x^{3}-8=0 $$

Short answer

The real solutions are x = 1 and x = -2.

Step-by-step solution

- Identify a substitution

Notice that the equation can be simplified using a substitution. Let’s set: y = x^3This transforms the original equation into a quadratic form.

- Substitute and simplify

Substitute y = x^3 into the given equation: y^2 + 7y - 8 = 0Now, solve the quadratic equation.

- Solve the quadratic equation

Factorize the quadratic equation: y^2 + 7y - 8 = 0 y^2 + 8y - y - 8 = 0 y(y + 8) - 1(y + 8) = 0 y = 1 or y = -8

- Substitute back to original variable

Substitute back y = x^3: x^3 = 1 or x^3 = -8

- Find the real solutions

Solve for x: x^3 = 1 gives x = 1 x^3 = -8 gives x = -2

Key concepts

Polynomial Substitution

Polynomial substitution is a useful technique for simplifying complex polynomial equations. It involves replacing a part of the polynomial with a new variable, making it easier to solve. For example, in the equation \(x^{6} + 7x^{3} - 8 = 0\), we notice that the term \(x^{6}\) can be expressed as \((x^{3})^{2}\). By setting \(y = x^{3}\), the original equation simplifies to \(y^{2} + 7y - 8 = 0\). This transformation makes the polynomial easier to handle, as we convert it into a quadratic form, which is simpler to solve.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form \(ax^{2} + bx + c = 0\). In our simplified equation, \(y^{2} + 7y - 8 = 0\), the coefficients are:

• a = 1
• b = 7
• c = -8

. Solving quadratic equations can be done using various methods such as factoring, completing the square, or the quadratic formula. For our example, we used factoring.

Real Solutions

Real solutions are the values of the variable that satisfy the equation and are real numbers (not imaginary or complex numbers). After simplifying our polynomial to a quadratic equation, we found the solutions to be \(y = 1\) and \(y = -8\). Since \(y\) represents \(x^{3}\), we then need to solve \(x^{3} = 1\) and \(x^{3} = -8\) to find the real solutions. For \(x^{3} = 1\), the real solution is \(x = 1\). For \(x^{3} = -8\), the real solution is \(x = -2\). Thus, the real solutions to the original equation \(x^{6} + 7x^{3} - 8 = 0\) are \(x = 1\) and \(x = -2\).

Factoring

Factoring is the process of breaking down a polynomial into simpler polynomials that, when multiplied together, give the original polynomial. For the quadratic equation \(y^{2} + 7y - 8 = 0\), we factorize it by expressing it as a product of two binomials: \(y^{2} + 7y - 8 = (y + 8)(y - 1) = 0\). Setting each factor equal to zero gives us the solutions:

• \(y + 8 = 0 \) thus \( y = -8 \)
• \(y - 1 = 0 \) thus \( y = 1 \)

. Factoring is a key technique in solving polynomial equations, especially when they can be transformed into quadratic equations, as shown in this problem.