Problem 22 - College Algebra with Corequisite Support

Question

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation. $$ x^{3}-3 x^{2}-10 x+24=0 $$

Step-by-Step Solution

Answer

The real solutions are \( x = 2, x = 4, x = -3 \).

Step 1: Identify the constant and leading coefficients

The Rational Zero Theorem states that any rational solution of the polynomial equation \( ax^n + bx^{n-1} + ... + k = 0 \)will be a factor of the constant term \( k \) divided by a factor of the leading coefficient \( a \).In this polynomial \( x^3 - 3x^2 - 10x + 24 = 0 \), the constant term is 24, and the leading coefficient is 1.

Step 2: List all potential rational zeros

According to the Rational Zero Theorem, the possible rational zeros are the factors of the constant term divided by the factors of the leading coefficient. Thus, the potential rational zeros are:Factors of 24: \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24 \) divided by 1.

Step 3: Use synthetic division to test rational zeros

To find the real zeros, use synthetic division to test each potential rational zero. Start with \( x = 1 \):- Coefficients: 1, -3, -10, 24.Perform synthetic division:1| 1 -3 -10 24 | 1 -2 -12 ---------------- 1 -2 -12 12Remainder: 12 (not zero, so 1 is not a solution).Continue testing other potential zeros with synthetic division.

Step 4: Find a valid rational zero

Continue testing potentials:For \( x = 2 \):- Coefficients: 1, -3, -10, 24.Perform synthetic division:2| 1 -3 -10 24 | 2 -2 -24 ---------------- 1 -1 -12 0Remainder is 0, therefore \( x = 2 \) is a solution.

Step 5: Factor the polynomial using the found zero

Since \( x = 2 \) is a zero, \( x - 2 \) is a factor. Using the quotient from the synthetic division of step 4, we have \( x^2 - x - 12 \).

Step 6: Solve the quadratic equation

Factor the quadratic equation \( x^2 - x - 12 = 0 \).Find factors: \( (x-4)(x+3) = 0 \).Set each factor to zero:\( x - 4 = 0 \) => \( x = 4 \)\( x + 3 = 0 \) => \( x = -3 \)

Step 7: List all real solutions

Combine all solutions found: From steps 4 and 6, the real solutions of the equation \( x^3 - 3x^2 - 10x + 24 = 0 \) are: \( x = 2, x = 4, x = -3 \).

Key Concepts

Synthetic Division

Synthetic Division is a method used to divide polynomials quickly, especially when checking potential zeros of a polynomial equation. In this method, you only need the coefficients of the polynomial, making it significantly faster and simpler than long division.
To perform synthetic division:

• Write down the coefficients of the polynomial you want to divide.
• Choose a potential zero (a candidate from the Rational Zero Theorem list).
• Perform division by bringing down the leading coefficient to the result line, then multiply it by the potential zero, and add to the next coefficient, repeating the process across the row of coefficients.

If the last entry in the bottom row (the remainder) is zero, the chosen potential zero is indeed a root of the polynomial. This not only helps in identifying zeros but also simplifies the polynomial, aiding in further factorization.

Polynomial Equation

A Polynomial Equation is an equation involving a polynomial expression. This means it consists of variables raised to varying powers, each multiplied by coefficients, summed up to potentially equate to zero.
For example, the polynomial equation in the exercise is:\[ x^3 - 3x^2 - 10x + 24 = 0 \]The powers of the variable, in this case, decrease from 3 all the way to 0. Solving polynomial equations involves finding the values of the variable (in this case, \( x \)) that satisfy the equation. Polynomial equations can have multiple solutions which can be real or complex, but we focus on the real solutions in this context as highlighted by the Rational Zero Theorem.

Factoring Polynomials

Factoring Polynomials is a crucial step in solving polynomial equations. Once a zero is found using methods like Synthetic Division, the polynomial can be divided by the corresponding linear factor, simplifying it into a polynomial of lower degree.
In the example, once you find \( x = 2 \) as a zero, the polynomial \( x^3 - 3x^2 - 10x + 24 \) can be factored using its quotient from synthetic division:

• The quotient \( x^2 - x - 12 \) is further factorable into \( (x-4)(x+3) \).

The complete factorization becomes \( (x-2)(x-4)(x+3) \), making it clear where all the solutions are (the root of each factor). This process not only simplifies solving but makes understanding the distribution of roots and how they relate to the original polynomial more intuitive.

Real Solutions

Real Solutions of a polynomial are the values of the variable, typically represented by \( x \), that satisfy the polynomial equation when substituted back into it, resulting in a true statement or zero.
From the fully factored form of our polynomial \( (x-2)(x-4)(x+3) \), we find the real solutions by setting each factor equal to zero:

• \( x-2 = 0 \rightarrow x = 2 \)
• \( x-4 = 0 \rightarrow x = 4 \)
• \( x+3 = 0 \rightarrow x = -3 \)

Thus, the real solutions of the equation \( x^3 - 3x^2 - 10x + 24 = 0 \) are \( x = 2, x = 4, \) and \( x = -3 \). Each of these solutions is a point at which the graph of the polynomial crosses the x-axis. These solutions are critical for understanding the polynomial's behavior and represent the x-values where the polynomial equals zero.