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Synthetic division is a simplified method used for dividing a polynomial by a binomial of the form \(x - c\). It is quicker and easier than the traditional polynomial long division. This method is particularly useful when testing potential rational zeros, which is precisely the situation we encounter in this polynomial exercise.To perform synthetic division, we arrange the coefficients of our polynomial in descending order of power. If we decide to test \(x = -1\) as a potential zero, we place this value beside our row of coefficients: 2, 15, 31, 20, and 4. We then bring down the leading coefficient (2 in this example), and multiply it by \(-1\), adding each result to the next coefficient in line.

• First box: Multiply 2 by \(-1\), yielding \(-2\), add to 15 to get 13.
• Second box: Multiply 13 by \(-1\), yielding \(-13\), add to 31 to get 18.
• Repeat this process until the last coefficient is reached.

If the final sum is zero, \(x = -1\) is a zero of the polynomial, as shown in our step-by-step process, proving \(x = -1\) and \(x = -\frac{1}{2}\) are both roots.

When a polynomial is reduced to a quadratic form, solving it is straightforward with the quadratic formula. The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula provides the roots of any quadratic equation \(ax^2 + bx + c = 0\). In our case, after factoring out \(x = -1\) and \(x = -\frac{1}{2}\), we are left with the quadratic \(2x^2 + 14x + 8\).Insertion of the coefficients \(a = 2\), \(b = 14\), and \(c = 8\) into the formula allows us to solve for the remaining zeros.

• Calculate the discriminant: \(b^2 - 4ac\). This value determines the nature of the roots.
• Plug the values into the quadratic formula to find the exact roots.

The computed roots are \(x = \frac{-7 + \sqrt{33}}{2}\) and \(x = \frac{-7 - \sqrt{33}}{2}\), which means our polynomial has two distinct irrational zeros.

Descartes' Rule of Signs is a wonderful tool for predicting the number of positive and negative real roots of a polynomial. You simply look at the changes in sign of the coefficients.For example, consider the terms in our polynomial \(P(x) = 2x^4 + 15x^3 + 31x^2 + 20x + 4\):

• All coefficients are positive, so there are no changes in sign.
• This implies there are no positive real roots according to Descartes' Rule.

To consider negative roots, change \(x\) to \(-x\) and examine the signs:

• The expression becomes \(2(-x)^4 - 15(-x)^3 + 31(-x)^2 - 20(-x) + 4\).
• Changes in coefficients' signs include: 0 to 1, indicating one negative root.

This aligns with the solutions found with synthetic division, confirming \(x = -1\) and \(x = -\frac{1}{2}\) as zeros.

Factoring is another essential technique in solving polynomials. It involves expressing the polynomial as a product of its simpler components. This is crucial when polynomial division leads to a quadratic expression.After using synthetic division to factor out zeros \(x = -1\) and \(x = -\frac{1}{2}\), we obtain a quadratic polynomial \(2x^2 + 14x + 8\).The strategies for factoring include:

• Looking for the greatest common factor (GCF) in expressions.
• Applying the quadratic formula when the quadratic is not readily factorable by other means.

In this exercise, the quadratic equation was not suited for simple factorization by inspection, so the quadratic formula was necessary to identify the remaining irrational roots. These factoring techniques allow you to break down complex expressions into manageable pieces, solving for all zeros efficiently.