Problem 36 - College Algebra with Corequisite Support

Question

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation. $$ 2 x^{4}-3 x^{3}-15 x^{2}+32 x-12=0 $$

Step-by-Step Solution

Answer

Use the Rational Zero Theorem to find and verify rational solutions such as \( x = 1 \) and other solutions by testing and dividing the polynomial.

Step 1: Identify the constant and leading coefficient

The Rational Zero Theorem tells us that the potential rational zeros of a polynomial equation are the factors of the constant term divided by the factors of the leading coefficient. For the polynomial \( 2x^{4} - 3x^{3} - 15x^{2} + 32x - 12 \), the constant term is \(-12\) and the leading coefficient is \(2\).

Step 2: List the potential rational zeros

List all factors of the constant term \(-12\): \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \). List all factors of the leading coefficient \(2\): \( \pm 1, \pm 2 \). The potential rational zeros are: \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{5}{2}, \pm \frac{11}{2} \).

Step 3: Test the potential rational zeros

Substitute each potential zero into the polynomial to check if it results in zero. Begin with \(x = 1\). Substitute into the polynomial: \[ 2(1)^4 - 3(1)^3 - 15(1)^2 + 32(1) - 12 = 2 - 3 - 15 + 32 - 12 = 4 \]Continue this process until you find values of \( x \) such that the polynomial equals zero.

Step 4: Confirm solutions using synthetic division

Once a potential zero is found (for instance, \( x = 1 \) does not work, but assume \( x = a \) works), use synthetic division to divide the polynomial by \( (x-a) \). Confirm the quotient is a polynomial of degree 3, which can be further analyzed to find more zeros. Continue this process for the other zeros found.

Step 5: Repeat for remaining zeros

Continue testing and using synthetic division on remaining possible rational zeros to find additional solutions until all solutions are found or exhausted.

Key Concepts

Polynomial Equation

A polynomial equation, such as the one we're examining, is an expression involving a polynomial set equal to zero. Here, our polynomial is given as \[ 2x^{4} - 3x^{3} - 15x^{2} + 32x - 12 = 0 \]The goal is to find the values of \( x \) that satisfy this equation, meaning they make the whole expression equal to zero. These particular values are known as the roots or zeros of the polynomial. Here, we specifically aim to find the real solutions using the Rational Zero Theorem. Equations like this can have one or multiple real or complex solutions or even none in the case where no real solutions exist.

Real Solutions

Real solutions refer to the values of \( x \) that solve the equation and are real numbers. In other words, these are the values you could plot on a number line. When dealing with polynomials, especially of higher degrees such as the fourth degree, as in this exercise, there can often be more than one real solution, or none at all if the solutions are complex. To find these real solutions, we first list and test the potential zeros using rational numbers derived from the Rational Zero Theorem process. After potential zeros are confirmed through substitution into the polynomial, they must result in the entire equation equating to zero if they are true real solutions.

Synthetic Division

Synthetic division is a simplified process of dividing a polynomial by a linear binomial of the form \( x - a \). This method is essential for verifying whether a given number is indeed a root of the equation. Here's a step-by-step way to understand synthetic division:

• Write down the coefficients of the polynomial — for our equation, they are \( 2, -3, -15, 32, -12 \).
• Choose a potential zero \( a \) and write it to the left of your coefficients.
• Bring down the first coefficient as is.
• Multiply this coefficient by \( a \), place the result under the second coefficient, and add them.
• Repeat the process — multiply the result by \( a \), add to the next coefficient — working your way through the list.
• A remainder of zero confirms \( a \) is a root, and the other numbers represent the new coefficients for a reduced polynomial.

This efficient technique helps verify zeros and simplify the polynomial degree by one with each successful division.

Potential Rational Zeros

Potential rational zeros are the possible values derived from the Rational Zero Theorem that could be solutions for the polynomial equation. These values are found as the possible fractions formed by the divisors of the constant term and the leading coefficient of the polynomial. For the polynomial \( 2x^{4} - 3x^{3} - 15x^{2} + 32x - 12 \), the potential rational zeros can be written as:\[ \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{5}{2}, \pm \frac{11}{2} \]These are tested by substituting into the polynomial to determine if they make the expression zero. Confirming a real zero leads to using synthetic division for simplification and further deduction of additional zeros. This process continues until all possible real solutions are found.