Problem 32 - Algebra and Trigonometry

Question

Find all rational zeros of the polynomial. $$ P(x)=8 x^{3}+10 x^{2}-x-3 $$

Step-by-Step Solution

Answer

The rational zeros are \( x = -1, \frac{1}{2}, -\frac{3}{4} \).

Step 1: Identify possible rational zeros using the Rational Root Theorem

The Rational Root Theorem states that potential rational zeros of a polynomial equation \( P(x) \) are given by \( \pm \frac{p}{q} \), where \( p \) is a factor of the constant term (-3) and \( q \) is a factor of the leading coefficient (8). The factors of -3 are \( \pm 1, \pm 3 \) and the factors of 8 are \( \pm 1, \pm 2, \pm 4, \pm 8 \). This gives potential zeros: \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{8}, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm \frac{3}{8} \).

Step 2: Test potential zeros using synthetic division

Test each potential zero by substituting it into \( P(x) \) using synthetic division or direct evaluation to find which values give a remainder of zero. Begin with \( x = 1 \) and evaluate: \( 8(1)^3 + 10(1)^2 - 1 - 3 = 8 + 10 - 1 - 3 = 14 \). Since it is not zero, \( x = 1 \) is not a root. Repeat this process for other potential zeros until one results in a zero remainder.

Step 3: Find actual zeros

Continue testing the potential zeros and find that \( x = -1 \) works: \( 8(-1)^3 + 10(-1)^2 - (-1) - 3 = -8 + 10 + 1 - 3 = 0 \). Thus, \( x = -1 \) is a zero. Using synthetic division or polynomial division, divide \( P(x) \) by \( x + 1 \) to find the remaining factor.

Step 4: Solve the reduced polynomial

The division of \( P(x) \) by \( x + 1 \) yields a quadratic \( 8x^2 + 2x - 3 \). Solve \( 8x^2 + 2x - 3 = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 8, b = 2, c = -3 \). Calculating the discriminant gives \( b^2 - 4ac = 4 + 96 = 100 \). So, \( x = \frac{-2 \pm 10}{16} \), yielding roots \( x = \frac{1}{2} \) and \( x = -\frac{3}{4} \).

Step 5: Verify the zeros found

Verify that the zeros \( x = -1, x = \frac{1}{2}, x = -\frac{3}{4} \) satisfy the polynomial by substitution. Recomputing \( P(-1), P(\frac{1}{2}), \text{ and } P(-\frac{3}{4}) \) shows all result in zero, confirming they are correct.

Key Concepts

Rational Root Theorem

Discovering possible zeros of a polynomial is made simpler by the Rational Root Theorem. This theorem is a powerful tool that gives you a set of potential rational zeros based on the polynomial's coefficients. Here's how it works:
The potential rational zeros of a polynomial with integer coefficients are found using the formula \( \pm \frac{p}{q} \), where:

• \( p \) is a factor of the constant term (the term without an \( x \)) of the polynomial.
• \( q \) is a factor of the leading coefficient (the coefficient of the highest power of \( x \)).

In our original exercise, the polynomial is \( P(x) = 8x^3 + 10x^2 - x - 3 \). The constant term is \(-3\), and the leading coefficient is \(8\). This gives us potential rational zeros of \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{8}, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm \frac{3}{8} \).
While not all of these values are actual zeros, they narrow down the possibilities, making it easier to find true roots.

Synthetic Division

After identifying possible zeros through the Rational Root Theorem, the next step is to test them. This is where Synthetic Division comes in handy.
Synthetic Division is a quick and easy method to divide a polynomial by a binomial of the form \( x - c \). Instead of performing long polynomial division, this method allows you to verify if a potential zero is indeed a zero by checking if it yields a remainder of zero.
For example, when we tested \( x = -1 \) in our polynomial \( P(x) \), Synthetic Division shows:

• Write down the coefficients of the polynomial: \( 8, 10, -1, -3 \).
• Bring down the leading coefficient as it is: \( 8 \).
• Multiply this leading number by \(-1\) and add to the next coefficient: \( 10 + (-8) = 2 \).
• Continue this process through all coefficients: \( 2 + (-2) = 0 \), confirming no remainder is left.

This process shows that \( x = -1 \) is a valid zero of the polynomial.

Quadratic Formula

Once you have found a zero using Synthetic Division, often, the polynomial reduces to a simpler form like a quadratic. At this point, you can use the Quadratic Formula to find the other zeros.
The Quadratic Formula is essential for solving any quadratic equation of the form \( ax^2 + bx + c = 0 \). The formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our situation, the polynomial was reduced to \( 8x^2 + 2x - 3 \). By plugging in \( a = 8, b = 2, \) and \( c = -3 \) into the formula, we calculate:

• The Discriminant: \( b^2 - 4ac = 4 + 96 = 100 \).
• Using the formula: \( x = \frac{-2 \pm \sqrt{100}}{16} \).
• This yields \( x = \frac{1}{2} \) and \( x = -\frac{3}{4} \).

This confirms two additional zeros of the polynomial.

Polynomial Division

Polynomial Division helps in dividing one polynomial by another polynomial, simplifying complex expressions to facilitate finding polynomial roots. In solving polynomial zeros, we often employ Polynomial Division after determining an initial zero using methods like Synthetic Division.
For instance, after verifying that \( x = -1 \) is a zero, you can break down the polynomial further through division. Let’s see how it works in practice:
Given \( P(x) = 8x^3 + 10x^2 - x - 3 \) is initially divided by \( x + 1 \), the known zero's factor:

• Setup division: Place the divisor \( x + 1 \) and align it with the polynomial.
• Divide the leading term of the polynomial by the leading term of the divisor and write the result.
• Multiply and subtract, then bring down the next term.
• Repeat until all terms are exhausted.

The division results in \( 8x^2 + 2x - 3 \), a simpler quadratic expression, making it easy to apply the Quadratic Formula subsequently. After finding roots, confirm them by substituting back into the original equation to ensure they satisfy it.