Problem 38 - College Algebra with Corequisite Support

Question

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation. $$ 4 x^{3}-3 x+1=0 $$

Step-by-Step Solution

Answer

The real solutions are \(x = -1\) and \(x = \frac{1}{2}\).

Step 1: Apply Rational Zero Theorem

The Rational Zero Theorem suggests that any possible rational solution of the polynomial equation \(4x^3 - 3x + 1 = 0\) is of the form \(\frac{p}{q}\), where \(p\) is a factor of the constant term (the term not involving \(x\), which is 1) and \(q\) is a factor of the leading coefficient (the leading term 4). This gives us potential rational zeroes: \(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}\).

Step 2: Test Potential Rational Zeros

Substitute each potential zero into the polynomial to check if it makes the equation true (i.e., evaluates to zero). For example, test \(x = 1\). Substitute into the polynomial: \[ 4(1)^3 - 3(1) + 1 = 2 \] This does not equal zero, so \(x = 1\) is not a root. Next, test \(x = -1\): \[ 4(-1)^3 - 3(-1) + 1 = 0 \] This evaluates to zero, so \(x = -1\) is a root.

Step 3: Factor the Polynomial Using Discovered Zero

Since \(x = -1\) is a root, \(x + 1\) is a factor of the polynomial. Use polynomial division or synthetic division to divide the polynomial \(4x^3 - 3x + 1\) by \(x + 1\). The quotient is \(4x^2 - 4x + 1\).

Step 4: Solve the Quadratic Equation

The equation \(4x^2 - 4x + 1 = 0\) can be solved using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 4\), \(b = -4\), and \(c = 1\). Substituting these into the formula: \[ x = \frac{4 \pm \sqrt{(-4)^2 - 4(4)(1)}}{8} = \frac{4 \pm \sqrt{0}}{8} = \frac{4}{8} = \frac{1}{2} \] Thus, the roots of the quadratic equation are equal: \(x = \frac{1}{2}\).

Step 5: Compile All Real Solutions

The real solutions to the original polynomial \(4x^3 - 3x + 1 = 0\) are therefore \(x = -1\) and \(x = \frac{1}{2}\).

Key Concepts

Polynomial Division

Polynomial division is a crucial technique used in algebra to break down polynomials into simpler components. Much like dividing numbers, polynomial division involves dividing a polynomial, called the dividend, by another polynomial, known as the divisor. The goal is to find a quotient polynomial and sometimes a remainder. There are two primary methods for this: long division and synthetic division. Each method provides a structured way to divide, helping identify factors and solutions for polynomial equations.

• Dividend: The polynomial being divided.
• Divisor: The polynomial dividing the dividend.
• Quotient: The result of the division, representing how many times the divisor fits into the dividend.

Understanding polynomial division is fundamental when it comes to factoring and solving polynomial equations, as it allows you to simplify complex expressions.

Synthetic Division

Synthetic division is a shortcut method to divide polynomials, specifically when dividing by linear polynomials of the form \(x - c\). This method is more efficient than long division and is particularly useful when evaluating potential rational zeros provided by the Rational Zero Theorem. To perform synthetic division:

• Write down the coefficients of the dividend polynomial.
• Use the zero of the divisor (e.g., \(x + 1\) means the zero is \(-1\)).
• Follow the synthetic division process of bring down, multiply, and add to determine the coefficients of the quotient and the remainder.

Despite its simplicity compared to traditional long division, mastering synthetic division requires practice to ensure accuracy and efficiency. It often helps locate roots of polynomials and plays a key role in simplifying expressions before solving further.

Quadratic Formula

The quadratic formula is a universal solution for finding roots of quadratic equations of the form \(ax^2 + bx + c = 0\). Given the coefficients \(a\), \(b\), and \(c\), the formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula derives all possible solutions, including real and complex solutions, based on the discriminant \(b^2 - 4ac\). When the discriminant:

• Is positive, two distinct real solutions exist.
• Is zero, one unique real solution exists (also known as a repeated root).
• Is negative, no real solutions exist, resulting in two complex solutions.

The quadratic formula is a powerful tool for solving any quadratic equation and is an essential part of algebra and pre-calculus curricula.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. This process is akin to finding common denominators or breaking down numbers into prime factors. Factoring is important because it reveals the roots of the polynomial and simplifies problems involving polynomial expressions.Types of factoring include:

• Factoring by Grouping: Splitting terms into groups that have common factors.
• Using the Difference of Squares: Identifying terms that form \(a^2 - b^2 = (a - b)(a + b)\).
• Recognizing Perfect Square Trinomials: Expressions that can be rewritten as squared binomials.
• Simple Factor Extraction: Taking out the greatest common factor from all terms.

Through factoring, complex polynomial equations are broken down into solvable forms, allowing easier identification of solutions and simplifying mathematical analysis.