Problem 60 - Precalculus Mathematics for Calculus

Question

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(a)\) $$P(x)=3 x^{3}+17 x^{2}+21 x-9$$

Step-by-Step Solution

Answer

The real zeros of the polynomial are \( x = -3, \frac{-4 + \sqrt{7}}{3}, \) and \( \frac{-4 - \sqrt{7}}{3} \).

Step 1: Look for any obvious integer roots using the Rational Root Theorem

To check for any rational integer roots, the Rational Root Theorem suggests potential roots are the factors of the constant term (-9) over the factors of the leading coefficient (3). Potential factors are \( \pm 1, \pm 3, \pm 9 \). Test these possible roots by substituting into the polynomial.

Step 2: Substitute the potential roots into P(x) to check for actual roots

Begin by substituting \( x = 1 \) into \( P(x) = 3x^3 + 17x^2 + 21x - 9 \): \( P(1) = 3(1)^3 + 17(1)^2 + 21(1) - 9 = 32 \, \text{(not a root)} \). Substitute \( x = -1 \): \( P(-1) = 3(-1)^3 + 17(-1)^2 + 21(-1) - 9 = -14 \, \text{(not a root)} \). Continue testing other potentials until identifying a root.

Step 3: Identify a real root

Upon substituting \( x = -3 \):\( P(-3) = 3(-3)^3 + 17(-3)^2 + 21(-3) - 9 = 0 \) This shows \( x = -3 \) is a root of the polynomial.

Step 4: Use synthetic division to divide the polynomial

Since \( x = -3 \) is a root, use synthetic division to divide \( P(x) \) by \( x + 3 \). Perform synthetic division on \( 3x^3 + 17x^2 + 21x - 9 \) by \( x + 3 \). The result of the division provides a quadratic polynomial.

Step 5: Write the polynomial as a product of a linear and a quadratic polynomial

The quotient from division is \( 3x^2 + 8x + 3 \). Thus, we have \( P(x) = (x + 3)(3x^2 + 8x + 3) \).

Step 6: Solve the quadratic equation for remaining roots

Use the quadratic formula on the equation \( 3x^2 + 8x + 3 = 0 \):\[ x = \frac{-8 \pm \sqrt{8^2 - 4 \times 3 \times 3}}{2 \times 3} \] Calculate the discriminant: \( 64 - 36 = 28 \). Thus, the equation becomes \[ x = \frac{-8 \pm \sqrt{28}}{6} \].

Step 7: Simplify the solutions from the quadratic formula

Simplify \( \sqrt{28} = 2\sqrt{7} \). So, the solutions are \[ x = \frac{-8 \pm 2\sqrt{7}}{6} \]. Further simplification yields: \[ x = \frac{-4 \pm \sqrt{7}}{3} \]. Thus, the two additional real roots are \( \frac{-4 + \sqrt{7}}{3} \) and \( \frac{-4 - \sqrt{7}}{3} \).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a powerful tool for discovering rational solutions, or roots, of a polynomial equation. When you're investigating a polynomial like \( P(x) = 3x^3 + 17x^2 + 21x - 9 \), this theorem helps you identify potential rational solutions by using the ratio of factors. Here's how it works:

• Identify the constant term and the leading coefficient of the polynomial.
• In our case, the constant term is \(-9\) and the leading coefficient is \(3\).
• List all factors of the constant term \(-9\): \(\pm 1, \pm 3, \pm 9\).
• List all factors of the leading coefficient \(3\): \(\pm 1, \pm 3\).
• The potential rational roots are obtained by forming all possible fractions with these factors, which in this case simplifies to the same list \(\pm 1, \pm 3, \pm 9\).

To find a root, substitute these values into the polynomial to determine whether they yield zero. This method helps narrow down potential candidates for real roots, making it easier to solve the polynomial equation.

synthetic division

Synthetic division is a streamlined method of dividing polynomials, especially when you have already found one root. It's much quicker and less cumbersome than traditional polynomial long division. Here's a quick guide to understanding how it works, using the example where we found \( x = -3 \) is a root of the polynomial:

• Write down the coefficients of the polynomial. For \( 3x^3 + 17x^2 + 21x - 9 \), you have: \(3, 17, 21, -9\).
• Place the root (in this case, \(-3\)) outside the synthetic division box.
• Bring down the first coefficient (\(3\)) directly as it is.
• Multiply this number by the root \(-3\) and write the result below the second coefficient.
• Add the two values and write down the sum. Repeat multiplication and addition until you complete the table.
• The final row gives you the coefficients of the quotient polynomial.

When you perform synthetic division for this polynomial by \( x + 3 \), you end up with a simpler quadratic \( 3x^2 + 8x + 3 \). This becomes the focus for finding additional roots.

quadratic formula

The quadratic formula is a universal tool for finding the roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \). When your quadratic can't be easily factored, turn to this formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Let's apply it to our quotient polynomial from the problem, \(3x^2 + 8x + 3 = 0\):

• First, calculate the discriminant \(b^2 - 4ac\). Here, \(b = 8\), \(a = 3\), and \(c = 3\), so \(64 - 36 = 28\).
• The discriminant (28) is positive, indicating two distinct real roots.
• Substitute back into the formula: \( x = \frac{-8 \pm \sqrt{28}}{6} \).
• Simplify \(\sqrt{28}\) to \(2\sqrt{7}\), and simplify the fraction: \( x = \frac{-4 \pm \sqrt{7}}{3}\).

This procedure delivers two precious real zeros of the polynomial, supplementing the solution obtained through rational root testing and synthetic division.