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The Rational Root Theorem is a handy tool for finding possible rational zeros of a polynomial. It states that any rational solution, or root, of a polynomial equation with integer coefficients, is a fraction \( \frac{p}{q} \), where **p** is a factor of the constant term and **q** is a factor of the leading coefficient. This theorem helps us narrow down the list of values to test when looking for roots. In the example polynomial \( P(x)=2x^3-8x^2+9x-9 \), we use this theorem to find that the potential rational roots are:

• \( \pm 1, \pm 3, \pm 9 \)
• \( \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{9}{2} \)

These possible roots are based on the factors of the constant term \(-9\) and the leading coefficient \(2\). This theorem dramatically reduces the number of possible roots to check.

Synthetic division is a simplified method of dividing polynomials, especially useful when evaluating a polynomial at a given value or when checking potential roots. It is more efficient than traditional long division for polynomials, particularly when the divisor is of the form \( x - c \). Here is how it works:- Write down only the coefficients of the polynomial.- Bring down the leading coefficient.- Multiply it by the proposed root, then add it to the next coefficient.- Repeat this process until the last number.In our example, testing \( x = 3 \) as a root, we start with the coefficients \( 2, -8, 9, -9 \). After performing the synthetic division, we verify that \( x = 3 \) is indeed a root because the remainder is zero. This confirmation allows us to factor \( (x - 3) \) from the polynomial.

When a polynomial has been simplified to a quadratic form, the quadratic formula can be used to find its roots. The formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula calculates the roots for the quadratic equation \( ax^2 + bx + c = 0 \). It's especially useful when the equation doesn't factor easily. In the reduced polynomial from our exercise, \( 2x^2 - 2x + 3 = 0 \), we apply the quadratic formula. Here, \( a = 2 \), \( b = -2 \), and \( c = 3 \). The discriminant, \( b^2 - 4ac \), is calculated as \(-20\), which is negative, indicating complex roots.

Complex roots occur when the discriminant in the quadratic formula is negative, meaning the square root of a negative number is needed. This introduces imaginary numbers, represented by \( i \), where \( i = \sqrt{-1} \). In our exercise, the discriminant of \( 2x^2 - 2x + 3 \) was \(-20\), so the complex roots are found using:\[ x = \frac{2 \pm \sqrt{-20}}{4} = \frac{2 \pm 2\sqrt{5}i}{4} = \frac{1 \pm \sqrt{5}i}{2} \]These solutions, \( \frac{1 + \sqrt{5}i}{2} \) and \( \frac{1 - \sqrt{5}i}{2} \), are the complex roots of the quadratic part of the polynomial. They reveal that not all zeros of a polynomial need to be real numbers. Embracing complex roots expands the scope of solutions to any polynomial equation.