Problem 57 - Precalculus Mathematics for Calculus

Question

Find all zeros of the polynomial. $$P(x)=x^{4}-6 x^{3}+13 x^{2}-24 x+36$$

Step-by-Step Solution

Answer

Zeros are 1, 3, 1 + 3i, and 1 - 3i.

Step 1: Identify Polynomial Degree

The degree of the polynomial \(P(x) = x^4 - 6x^3 + 13x^2 - 24x + 36\) is 4. This indicates there are up to 4 real or complex roots to find.

Step 2: Use the Rational Root Theorem

The Rational Root Theorem suggests that any rational root, \(p/q\), is a factor of the constant term (36) divided by a factor of the leading coefficient (1). Thus, potential rational roots are ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36.

Step 3: Test Potential Rational Roots with Synthetic Division

Use synthetic division to test possible roots starting from the smallest absolute values. Upon testing, \(x = 1\) satisfies the polynomial, as there is no remainder:

Step 4: Perform Synthetic Division with x = 1

Divide the polynomial by \(x - 1\) using synthetic division. The quotient is \(x^3 - 5x^2 + 8x - 36\).

Step 5: Find Zeros of the Quotient Polynomial

Repeat the process starting with \(x^3 - 5x^2 + 8x - 36\). By testing roots, \(x = 3\) satisfies the quotient polynomial.

Step 6: Perform Synthetic Division with x = 3

Divide \(x^3 - 5x^2 + 8x - 36\) by \(x - 3\) using synthetic division. The quotient is \(x^2 - 2x + 12\).

Step 7: Find Zeros of Quadratic Polynomial

Solve \(x^2 - 2x + 12\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1, b = -2, c = 12\). This gives complex roots \(x = 1 \pm 3i\).

Step 8: Compile All Zeros

Combine all zeros from steps: \(x = 1\), \(x = 3\), and complex roots \(x = 1 + 3i, x = 1 - 3i\).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a powerful tool in algebra that helps in finding potential rational zeros of a polynomial. Here's how it works:

The theorem states that if a polynomial has a rational root \(\frac{p}{q}\), then \(p\) must be a factor of the constant term and \(q\) must be a factor of the leading coefficient. For the polynomial \(P(x) = x^4 - 6x^3 + 13x^2 - 24x + 36\), the constant term is 36, and the leading coefficient is 1.

• Factors of 36: ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36
• Factors of 1: ±1

Thus, our potential rational roots are all the factors of 36, which include ±1, ±2, ±3, etc. Once these candidates are identified, we can test them using the technique of synthetic division to confirm if they are actual roots.

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a linear binomial of the form \(x - c\). It is particularly helpful for checking the roots of a polynomial quickly.

Here's how synthetic division is performed:

1. Write down the coefficients of the polynomial.2. Identify the root you're testing (let's say \(x = 1\)).3. Perform the synthetic division process, which involves multiplying and adding, to see if the remainder is zero.

If you get a remainder of zero, then \(x = 1\) is a root of the polynomial. In our case, this step was used twice: first finding \(x = 1\), and then \(x = 3\), confirming that these values are indeed the roots.

Complex Roots

Complex roots arise when the discriminant of a quadratic equation (inside the square root of the quadratic formula) is negative. These are numbers that include a real part and an imaginary part.

For the quadratic \(x^2 - 2x + 12\), we used the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

With values \(a = 1\), \(b = -2\), and \(c = 12\), the discriminant \(b^2 - 4ac\) evaluates to \(-44\), which is negative. This indicates complex roots. The roots are calculated as:

\[ x = 1 \pm 3i \]

Here, "i" represents the imaginary unit, and these roots are parts of the solution we compiled at the end.

Quadratic Formula

The quadratic formula is a universal method used to find the roots of a quadratic equation \(ax^2 + bx + c = 0\). It can be used regardless of the type of roots: real or complex.

The formula is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

This formula takes into account the coefficients \(a\), \(b\), and \(c\) from the quadratic equation and calculates the roots based on the discriminant \(b^2 - 4ac\).

• If the discriminant is positive, the roots are real and different.
• If it is zero, roots are real and the same.
• If it's negative, the roots are complex.

This formula helped us find the complex roots of our polynomial problem in the last step. It is especially useful as it provides a direct computation when factoring is difficult or not apparent.