Problem 36 - Precalculus : Building Concepts and Connections

Question

Find all real solutions of the polynomial equation. $$3 x^{3}-7 x^{2}=-5 x+1$$

Step-by-Step Solution

Answer

The equation has two real roots, \(x = 1\) and \(x = 1/3\).

Step 1: Simplify the equation

Move all terms to one side of the equation to establish an equation of the form \(f(x) = 0\). To get this, add \(5x - 1\) to both sides of the given equation, which transforms it into: \[3x^3 - 7x^2 + 5x - 1 = 0\]

Step 2: Factorize the equation

Find a common root to simplify the equation into a linear term and a quadratic term. A helpful approach is to try for integer solutions first for cubic equations. Test \(x = 1\) by substituting it into the equation. If \(f(1) = 0\), that means \(x - 1\) is a factor of the equation. Doing this results in \(3*1^3 - 7*1^2 + 5*1 - 1 = 0\). So, the equation factors into: \[(x - 1)(3x^2 - 4x + 1) = 0\]

Step 3: Solve for x

Set each factor equal to zero and solve for \(x\), because any factor equaling zero will make the whole equation equal to zero. This gives three potential solutions: \(x = 1\), \(3x^2 - 4x + 1 = 0\). The first equation gives \(x = 1\) as a solution again. For the quadratic equation, use the quadratic formula: \(x = [-(-4) ± sqrt((-4)^2 - 4*3*1)] / (2*3)\) which simplifies to \(x = [4 ± sqrt(16 - 12)] / 6\) or \(x = [4 ± 2] / 6\). This gives two more solutions, \(x = 1\) and \(x = 1/3\).

Key Concepts

Cubic Equation

A cubic equation is a type of polynomial equation that features a term with an exponent of three, as its highest degree. The general form of a cubic equation can be represented as:

• \[ ax^3 + bx^2 + cx + d = 0 \]

where \(a, b, c,\) and \(d\) are coefficients, and \(a eq 0\) since that would make it a quadratic equation.
Solving cubic equations can be trickier than solving linear or quadratic equations. This is partly because they can have up to three real roots and partly due to the lack of a simple formula like the quadratic formula.

In practice, the process of solving cubic equations often involves techniques such as factoring or using numerical methods. Recognizing patterns or testing small integer roots can also be helpful. These methods allow us to simplify the equation and find its solutions step by step.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as the product of its factors. This is akin to expressing the number 12 as \(3 \times 4\) or \(2 \times 6\). For polynomials, especially cubic ones, the aim is to break down the polynomial into simpler terms which can be solved for values of \(x\).

To factor a cubic polynomial, determine if there is an obvious root by testing integer values such as \(-1, 0, 1, 2\), and so on. Once you find a root, use it to derive one of the polynomial’s linear factors, e.g., \(x - r\), where \(r\) is the root you found.

• For example, if \(f(1) = 0\), then \(x - 1\) is a factor.

Using this factor, you can then use polynomial division or synthetic division to factor the rest of the polynomial. In the end, this breaks it down to simpler quadratic or even linear parts that can be solved using other methods.

Factoring is particularly useful because any value that makes a factor zero will make the entire equation zero.

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of a quadratic equation. Quadratic equations are polynomials where the highest power of the variable is two. The standard form is:

• \[ ax^2 + bx + c = 0 \]

where \(a, b,\) and \(c\) are coefficients, and \(a eq 0\).
The quadratic formula is expressed as:

• \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

The symbol \(\pm\) means that there are typically two solutions to the quadratic equation.

This formula directly derives from the process of completing the square of the quadratic equation. It can always be used when factoring is not easily possible or apparent.
In the context of solving a cubic equation, once you've factored out a linear term, you might end up with a quadratic equation like \(3x^2 - 4x + 1 = 0\). Here, applying the quadratic formula helps find the remaining roots, providing you a complete solution to the original cubic equation.

Understanding and using the quadratic formula effectively is crucial for solving a wider range of polynomial equations.