Problem 21 - College Algebra

Question

a. List all possible rational roots. b. Use synthetic division to test the possible rational roots and find an actual root. c. Use the root from part (b) and solve the equation. $$ x^{4}-2 x^{3}-5 x^{2}+8 x+4=0 $$

Step-by-Step Solution

Answer

The only real root of the polynomial equation \(x^{4}-2 x^{3}-5 x^{2}+8 x+4=0\) is \(x=-1\), and the other three roots are non-real complex numbers.

Step 1: Identify Possible Rational Roots

By the Rational Root Theorem, if a polynomial has a rational root \(p/q\), where \(p\) is a factor of the constant term, and \(q\) is a factor of the leading coefficient, then the possible rational roots of the equation \(x^{4}-2 x^{3}-5 x^{2}+8 x+4=0\) are all the ratios \(p/q\) where \(p\) is an integer factor of 4 and \(q\) is an integer factor of 1. Specifically, they are \(\pm1, \pm2, \pm4\).

Step 2: Use Synthetic Division to Find an Actual Root

We use synthetic division to test each of the possible rational roots, starting from the smallest. After testing the possible roots, we find that -1 is a root of the polynomial, because synthetic division with -1 produces no remainder.

Step 3: Divide the Polynomial by the Root

Now, we divide the original polynomial by \(x+1\) (corresponding to the root \(x=-1\)), using either long division or synthetic division. This gives us a reduced polynomial \(x^{3}-3x^{2}-2x-4\).

Step 4: Solve the Reduced Polynomial

Now we need to solve the reduced polynomial. However, it's a cubic equation and also it doesn't factor nicely so none of the potential rational roots \(\pm1, \pm2,\pm4\) are roots of this cubic polynomial. Therefore, the original given polynomial has no rational roots, only one real root which is -1 and three other roots are non-real complex numbers.

Key Concepts

Rational Root Theorem

Understanding the Rational Root Theorem is crucial when dealing with polynomial equations. It is a mathematical principle that provides us with a set of potential rational solutions for any polynomial equation, like the one given \( x^{4}-2x^{3}-5x^{2}+8x+4=0 \).

According to the theorem, if a polynomial equation with integer coefficients has a rational solution \(\frac{p}{q}\), then 'p' must be a factor of the constant term (the term with no variable, in this case, 4), and 'q' must be a factor of the leading coefficient (the coefficient of the term with the highest exponent, in our example, 1 for \(x^{4}\)).

• The constant term factors: \(\pm1, \pm2, \pm4\).
• The leading coefficient factors: \(\pm1\).

To find all possible rational roots, we generate the combinations of these factors. This means the potential rational roots to check are \(\pm1, \pm2, \pm4\). The theorem narrows down the infinite possibilities to a finite number, making the task manageable. However, remember that these are just potential solutions. Further steps are needed to confirm whether any of them are actual solutions to the equation.

Synthetic Division

When it comes to actually testing the possible rational roots provided by the Rational Root Theorem, synthetic division is a handy tool. It is a streamlined form of polynomial division, particularly useful when dividing by a linear factor.

As applied to our equation \(x^{4}-2x^{3}-5x^{2}+8x+4=0\), we create a setup where the coefficients of the polynomial are written in descending order, and the potential root is placed to the left. For example, to test \(x = -1\), arrange the coefficients \(1, -2, -5, 8, 4\) accordingly and perform synthetic division:

• Bring down the leading coefficient.
• Multiply the potential root by the value just written down and place the result under the next coefficient.
• Add the numbers in the columns and continue the process.

If the last number, the remainder, is zero, the tested candidate is indeed a root. Using this method on \(x = -1\) for our given polynomial shows that it produces no remainder, confirming that -1 is an actual root of the polynomial.

Polynomial Roots

Once you identify a root using synthetic division, you can then focus on finding other polynomial roots. Knowing that -1 is a root of our example polynomial \( x^{4}-2x^{3}-5x^{2}+8x+4=0 \), we can then divide the polynomial by \(x+1\) to find the quotient polynomial. This quotient polishes our search area for the remaining roots.

Once you've divided the polynomial, you'll be left with a lower degree polynomial, which simplifies the problem. However, not every cubic or quadratic polynomial you arrive at will factor easily, or feature nice rational roots. In our example, the remaining cubic polynomial does not have rational roots, which means it must be solved with other methods, like numerical approximations, graphing, or using the cubic formula.

• If the reduced polynomial is quadratic, you can use the quadratic formula to find the remaining roots.
• For cubic or higher degree polynomials that don't factor readily, numerical methods or further algebraic manipulation may be required.

Identifying all roots of a polynomial may require combining numerous techniques and accepting that some roots could be irrational or complex.