Problem 52 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use synthetic division to show that \(x\) is a solution of the thirddegree polynomial equation, and use the result to factor the polynomial completely. List all the real solutions of the equation. Value of \(x\) \(x=4\) \(x=-6\) \(x=-\frac{3}{2}\) \(x=\frac{1}{3}\) \(x=\sqrt{3}\) \(x=2-\sqrt{5}\) Polynomial Equation $$x^{3}-x^{2}-13 x-3=0$$

Step-by-Step Solution

Answer

The roots of the equation \(x^{3}-x^{2}-13 x-3=0\) are \(x=4\) and \(x=-\frac{3}{2}\). The factorized form of the polynomial equation is \( (x-4)(x+1.5)(x+3) = 0 \)}.

Step 1: Understanding Synthetic division

Synthetic division is a shorthanded, less detailed way of dividing polynomials which helps to find out if a given \( x \) is a root of the polynomial in question. The coefficients of every term in the polynomial is the key focus in synthetic division.

Step 2: Conduct Synthetic Division for the given equation

Use the coefficients of the polynomial equation \(x^{3} - x^{2} - 13x -3 = 0\) which are [ 1, -1, -13, -3 ] and perform synthetic division with each given number for \( x \) to find out which one is the root.

Step 3: Results of Synthetic Division for each \( x \)

Apply synthetic division for each value of \( x \) mentioned. Find out which values of \( x \) give a remainder of zero. Ones that yield a zero remainder are roots of the given equation.

Step 4: List down all the real solutions

List all the real solutions or the value(s) of \( x \) that give a remainder of zero on synthetic division. Those are the real roots of the given polynomial equation.

Step 5: Factorize the Polynomial

Since synthetic division, in addition to confirming the given roots, also provides the reduced forms of the original polynomials, the factorized form can be written out directly. The original polynomial is the product of the linear form associated with the given root and the reduced polynomial derived from the synthetic division process.

Key Concepts

Polynomial Equations

Polynomial equations are mathematical expressions that include terms of variables raised to whole number powers and their coefficients. These are central to algebra and more advanced branches of mathematics. An equation like \(x^{3} - x^{2} - 13x - 3 = 0\) is an example, comprised of terms with the variable \(x\) raised up to the third power.

Understanding how to solve these equations is important because they can model many real-world scenarios. For example, predicting the trajectory of a ball thrown into the air can be represented by a polynomial equation. The goal when working with polynomial equations is to find the value or values of the variable that make the equation true, which is when the left side equals zero, indicating that these values are the roots of the polynomial.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler components, or 'factors', that when multiplied together give back the original polynomial. Simplifying the equation \(x^{3} - x^{2} - 13x - 3 = 0\) can make it more manageable and can reveal valuable information about the equation, such as its roots.

There are various methods of factoring, including pulling out a greatest common factor, grouping, and special patterns like difference of squares and sum/product of cubes. Factoring is a crucial step in simplifying polynomial equations to their most basic components in order to identify the solutions or graph the function more easily.

Roots of Polynomial

Identifying the roots of a polynomial is essentially finding the solutions to the equation formed by setting the polynomial to zero. Roots are the values of \(x\) that, when plugged into the polynomial, will cause the entire expression to equal zero.

In the context of the exercise, when using synthetic division with the given values of \(x\), we're searching for which values satisfy the equation and yield no remainder. These roots can be real numbers, complex numbers, or even repeating. Finding the roots is a fundamental process in solving polynomial equations, and the roots can tell us many things, such as the number of times a graph will touch or cross the x-axis at those points.

Real Solutions

Real solutions of polynomial equations are the 'x-intercepts' or roots that are real numbers, as opposed to complex or imaginary numbers. In the exercise, we're interested in finding these real solutions as they are often the most meaningful in real-world contexts.

After performing synthetic division and finding which values of \(x\) result in a zero remainder, these values are the real solutions. For instance, if \(x = 4\) leads to 'zero' remainder after synthetic division, then 4 is a real solution or real root of the polynomial. Real solutions are significant because they offer tangible conclusions in physical problems, like the height of a throw peaking at a certain time or the profitability of a business based on its production costs.