Problem 49 - Precalculus Mathematics for Calculus

Question

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(a)\). $$P(x)=x^{4}-6 x^{3}+4 x^{2}+15 x+4$$

Step-by-Step Solution

Answer

The real zeros are \( x = 1 \), \( x = -4 \), and \( x = \frac{1 \pm \sqrt{17}}{2} \).

Step 1: Analyze the Polynomial

The polynomial is of degree 4: \(P(x) = x^4 - 6x^3 + 4x^2 + 15x + 4\). Since it's a quartic polynomial, it may have up to four real zeros.

Step 2: Check for Rational Roots

Apply the Rational Root Theorem, which suggests that any rational root, in the form of \( \frac{p}{q} \), must be a factor of the constant term \( 4 \) divided by a factor of the leading coefficient \( 1 \). Thus, potential rational roots are \( \pm 1, \pm 2, \pm 4 \).

Step 3: Test Rational Roots with Synthetic Division

Test possible rational roots. Starting with \( x = 1 \) using synthetic division. Performing the division confirms that \( x - 1 \) is not a factor. Continue testing the rest: \( x = 2 \) and \( x = -1 \). Upon testing, \( x = 1 \) is a root.

Step 4: Factor the Polynomial

Divide \( P(x) \) by \( x - 1 \) using synthetic division to factor it. This results in a quotient polynomial: \( x^3 - 5x^2 - x + 4 \).

Step 5: Find Zeros of the Quotient Polynomial

Apply the Rational Root Theorem again to \( x^3 - 5x^2 - x + 4 \) with possible roots \( \pm 1, \pm 2, \pm 4 \). Use synthetic division again on this cubic polynomial and find that both \( x = 1 \) and \( x = -4 \) are zeros.

Step 6: Use Quadratic Formula

After factoring out \( (x - 1) \) and \( (x + 4) \), the remaining factor is \( x^2 - x - 4 \). Use the quadratic formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1, b = -1, c = -4 \), to solve the quadratic. The discriminant \( b^2 - 4ac = 1 + 16 = 17 \), thus the solutions are: \[ x = \frac{1 \pm \sqrt{17}}{2}. \]

Step 7: Compile the Real Zeros

The real zeros of the polynomial are gathered: \( x = 1 \) (twice, counting multiplicity), \( x = -4 \), and \( x = \frac{1 \pm \sqrt{17}}{2} \).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a powerful tool to explore potentially rational zeros of a polynomial. It helps us prioritize which potential roots to test by providing a list of possible rational candidates. For the polynomial \(P(x) = x^4 - 6x^3 + 4x^2 + 15x + 4\), we look for rational roots of the form \(\frac{p}{q}\). Here, \(p\) is a factor of the constant term (4), and \(q\) is a factor of the leading coefficient (1).
This simplifies our search to \(\pm 1, \pm 2,\) and \(\pm 4\). These are the possible rational roots we must test to find any zeros. Checking rational roots first can simplify a complex polynomial dramatically, sometimes leading to easier factors.

Synthetic Division

Synthetic division aids in polynomial division, allowing us to quickly validate and test potential roots from the Rational Root Theorem. It's a streamlined method that works effectively with polynomials divided by binomials of the form \((x - c)\).

For example, to verify if \(x = 1\) is a root of \(P(x) = x^4 - 6x^3 + 4x^2 + 15x + 4\), we perform synthetic division. The goal here is to divide the polynomial by \(x - 1\). If the result is a zero remainder, then \(x - 1\) is indeed a factor. Synthetic division decomposes a polynomial into a simpler form, aiding further analysis by reducing degree and complexity, leading to a quotient polynomial once a root is found.

Quadratic Formula

The quadratic formula is a widely-used algebraic tool that guarantees finding all roots of quadratic equations of the form \(ax^2 + bx + c = 0\). The formula itself is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula is critical when simpler factoring avenues are exhausted or unavailable. It systematically identifies real and complex roots.
In our example, after using synthetic division, we are left with the quadratic \(x^2 - x - 4\). Applying the quadratic formula with \(a = 1\), \(b = -1\), and \(c = -4\), the discriminant \(b^2 - 4ac\) turns out to be positive, indicating real roots. This precise step reveals the remaining zeros: \(x = \frac{1 \pm \sqrt{17}}{2}\), completing the discovery of all polynomial zeros.