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Synthetic division is a simplified form of dividing a polynomial by a binomial of the form \(x - c\). It is quicker and less cumbersome than long division. Here's how it works:

• Identify the coefficients of the polynomial you are dividing.
• Use the root of the divisor. For example, if you are dividing by \(3x + 2\), solve \(3x + 2 = 0\) to find \(x = -\frac{2}{3}\).
• List the coefficients: \(3, 2, -36, 24, 32\).
• Perform the synthetic division by using these coefficients and the divisor root.

For each step of the synthetic division:

• Multiply the root by the first coefficient and add it to the next.
• Repeat the process for all coefficients.
• The last number is the remainder. If it’s 0, you have factored correctly.

In our exercise, synthetic division resulted in a quotient of \(3x^{3} - 2x^{2} - 34x + 16\). Remember, synthetic division is an essential tool for polynomial factorization, especially when simplifying expressions becomes complex.

The Rational Root Theorem helps in finding the possible rational roots of a polynomial equation. This theorem suggests that if a polynomial has a rational root \(\frac{p}{q}\), then \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient. For example, consider the polynomial \(3x^{3} - 2x^{2} - 34x + 16\):

• The constant term is 16, and its factors are \(\pm 1, \pm 2, \pm 4, \pm 8, \pm 16\).
• The leading coefficient is 3, and its factors are \(\pm 1, \pm 3\).

Using the Rational Root Theorem, the potential rational roots are taken from all combinations of \(\frac{p}{q}\). Knowing all potential roots can significantly help in simplifying higher-degree polynomials through strategic testing and via synthetic division or direct substitution.In our exercise, testing led to discovering that \(x = 2\) is a root, helping factor the polynomial further.

When a polynomial expression reduces to a quadratic polynomial, the quadratic formula is an essential tool for factorization. The quadratic formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Where \(a\), \(b\), and \(c\) are coefficients from the quadratic expression \(ax^2 + bx + c\). This formula finds the roots of the quadratic equation, which are the values of \(x\) where the polynomial equals zero.In the given exercise, the quadratic \(3x^2 + 4x - 8\) was solved using the quadratic formula:

• \(a = 3, b = 4, c = -8\)
• Plugging into the formula yields roots \(x = \frac{-4 \pm 8}{6}\).

From these roots, the polynomial was further factored into \((3x - 2)(x + 4)\). Normally, the quadratic formula is your go-to method when other methods like factoring by grouping are too complex.

Polynomial division, both synthetic and long division, is used to divide polynomials when factorizing or simplifying expressions. It operates much like arithmetic long division.For a division \(\frac{f(x)}{d(x)}\):

• The polynomial \(f(x)\) is divided by a simpler polynomial \(d(x)\).
• Long division results in a quotient and sometimes a remainder.

Unlike synthetic division, regular polynomial (or long) division can handle any polynomials (not just linear). It's step-by-step where

• We divide the leading terms, multiply and subtract successively just as we do with numbers.
• Repeat for each descending power of the polynomial.

In our exercise, polynomial division started the factorization process and identified the resulting cubic polynomial. It's a fundamental technique in algebra that forms the backbone of handling polynomial expressions, providing structure to mathematical operations.