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In algebra, the Rational Root Theorem helps us possibly narrow down the rational roots for any polynomial with integer coefficients. It gives us a way to list out potential rational solutions by using the relationship between the constant term and the leading coefficient of the polynomial.

The theorem states that if \( \frac{p}{q} \) is a potential rational root (in its simplest form), \( p \) (numerator) must be a factor of the constant term, and \( q \) (denominator) must be a factor of the leading coefficient. For example, in the polynomial \( 6x^3 - 13x^2 + x + 2\), the constant term is 2 and the leading coefficient is 6. This means:

• Possible values for \( p \) (factors of 2): \( \pm 1, \pm 2 \).
• Possible values for \( q \) (factors of 6): \( \pm 1, \pm 2, \pm 3, \pm 6 \).

Combining these, the potential rational roots are: \( \pm 1, \pm 2, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{1}{6} \).

By checking these values, we can figure out which, if any, is a root of the polynomial.

Synthetic division is a simplified form of polynomial division, especially useful for testing potential roots derived from the Rational Root Theorem. It is quicker and less cluttered than the traditional long division method.

To use synthetic division, you line up the coefficients of the polynomial. For each potential root, you perform a series of operations, adjusting the coefficients step by step while testing division success. If the remainder is zero at the end, the tested number is indeed a root of the polynomial.

• Step 1: Write down the coefficients of the polynomial: 6, -13, 1, and 2.
• Step 2: Arrange the possible root, say 2, and perform calculation: bring down the first coefficient, multiply by the root, add to the next coefficient, and repeat.
• Step 3: If the last number (remainder) is zero, then the tested root is a correct rational root.

In our polynomial, testing shows that 2 is a root, resulting in a division that yields \(6x^2 - 7x - 1\).

Now, we look further to solve the resulting polynomial using additional methods.

When a polynomial is reduced to a quadratic form after identifying a rational root, the quadratic formula becomes a highly effective tool. It provides the exact roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \).

The formula is given by:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This works for quadratics obtained through methods like synthetic division. In our example, the quadratic equation formed is \( 6x^2 - 7x - 1 = 0 \). Here, \( a = 6 \), \( b = -7 \), and \( c = -1 \).

Plugging these into the quadratic formula:

• First, calculate \( b^2 - 4ac \): \( (-7)^2 - 4(6)(-1) = 49 + 24 = 73 \).
• Determine the roots: \( x = \frac{7 \pm \sqrt{73}}{12} \).

These provide us two additional roots, apart from the rational ones already identified.