Problem 37 - College Algebra

Question

Use synthetic division to divide \(f(x)=x^{3}-4 x^{2}+\) \(x+6\) by \(x+1 .\) Use the result to find all zeros of \(f\).

Step-by-Step Solution

Answer

The zeros of the function \(f(x)\) are \(x=-1\), \(x=\frac{3 + i\sqrt{7}}{2}\), and \(x=\frac{3 - i\sqrt{7}}{2}\)

Step 1: Synthetic Division

First, write down the coefficients of \(f(x)\) : 1 -4 1 6. The root corresponds to \(x+1\), so use -1 for the synthetic division. Set-up the synthetic division as follows: Under the horizontal line write the first number of the coefficients (1). After that, multiply -1 by the number you just wrote down, place this product under the next number up top and add to find the new number on the bottom (-4+1=-3). Repeat this process (-1*(-3)=3, 1+3=4, -1*4=-4, 6-4=2), thus obtaining the new polynomial \(x^2 - 3x + 4\).

Step 2: Finding the Remainder

The final bottom number (2) is the remainder. Note that for synthetic division, the result is a polynomial whose degree is one lower than the original polynomial. Hence, the polynomial from the result of the synthetic division is: \(x^2 - 3x + 4\).

Step 3: Find Zeros of the Polynomial

Set \(x^2 - 3x + 4\) equal to zero and solve for \(x\). Discadia's formula yields: \( x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(4)}}{2(1)} \), which simplifies to \( x = \frac{3 \pm \sqrt{-7}}{2} \). Therefore, the solutions are \(x=-1\), \(x=\frac{3 + i\sqrt{7}}{2}\), and \(x=\frac{3 - i\sqrt{7}}{2}\).

Key Concepts

Polynomial Division

Polynomial division is a method used to divide polynomials, much like how we divide numbers. When dividing a polynomial by a linear divisor like \(x + c\), synthetic division is a simplified alternative to the long division of polynomials.

If you were to use synthetic division, you focus on the coefficients of the polynomial and the zero of the divisor. For instance, given a polynomial \(f(x) = x^3 - 4x^2 + x + 6\), and you want to divide by \(x + 1\), you'd set \(x + 1 = 0\) to solve for \(x\), resulting in \(x = -1\).

Next, use synthetic division by working with the coefficients [1, -4, 1, 6] and our chosen \(-1\). Simply follow a pattern of multiplying and adding to find your result. This gives you a new polynomial of one less degree: \(x^2 - 3x + 4\).

Key takeaways include:

• It simplifies handling polynomials.
• Reduces polynomial's degree by one.
• Quickly finds factors and remainders.

Zeros of Polynomial

Finding the zeros of a polynomial is crucial because they represent the solutions of the equation when set to zero. The zeros are where the graph of the polynomial crosses the x-axis.

From the synthetic division of \(f(x) = x^3 - 4x^2 + x + 6\), we get a simpler polynomial \(x^2 - 3x + 4\). To find the zeros, set it equal to zero: \(x^2 - 3x + 4 = 0\).

Sometimes, zeros can be found using factoring methods, but when that isn't straightforward, more advanced methods like the quadratic formula come in handy. In this instance, it shows that the polynomial doesn't cross real points on the x-axis, indicating complex solutions.

Key insights:

• Zeros = solutions of the polynomial equation.
• Can be real or complex numbers.
• Essential in graphing polynomial functions.

Quadratic Formula

The quadratic formula is a powerful tool for finding solutions (zeros) of quadratic equations of the form \(ax^2 + bx + c = 0\). If direct factoring is difficult or impossible, the quadratic formula will compute the zeros.

The formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For the polynomial \(x^2 - 3x + 4\), \(a = 1\), \(b = -3\), and \(c = 4\). Substituting these values into the quadratic formula yields complex solutions since \(b^2 - 4ac\) gives a negative result: \( \sqrt{-7} \).

This leads to zeros \(x = \frac{3 \pm i\sqrt{7}}{2}\), where \(i\) represents the imaginary unit. With this, we understand that the zeros are not intercepts on the real plane but lie on the complex plane.

Takeaways:

• Useful for finding non-factorable roots.
• Handles complex and real solutions.
• Illuminates solutions that aren't apparent from inspection alone.