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The Rational Root Theorem is a useful tool in solving polynomial equations. It helps in identifying potential rational roots of a polynomial. To apply this theorem, we look at the constant term and the leading coefficient of the polynomial. For instance, in the given polynomial \(3x^3 + 11x^2 + 5x - 3\), the constant term is \(-3\) and the leading coefficient is \(3\). The theorem states that any potential rational root is of the form \(\frac{p}{q}\), where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient.

• Here, the factors of \(-3\) are \(\pm 1, \pm 3\).
• The factors of \(3\) are \(\pm 1, \pm 3\).

By dividing these factors, we find potential rational roots: \(\pm 1, \pm 3, \pm \frac{1}{3}, \pm \frac{3}{3}\). Naturally, \(\pm \frac{3}{3}\) simplify to \(\pm 1\). So, you only need to test \(\pm 1, \pm 3, \pm \frac{1}{3}\). These are not guaranteed to be roots, but they provide a valuable starting point in solving the equation without a calculator.

Synthetic Division is a simplified form of polynomial division, especially useful when testing potential roots of a polynomial. It is much quicker and easier than long division. When using synthetic division, you deal directly with the coefficients of the polynomial.For example, take the polynomial \(3x^3 + 11x^2 + 5x - 3\), and test if \(x = 1\) could be a root. We list down the coefficients: \(3, 11, 5, -3\). Using \(x = 1\) (or any other potential root), synthetic division helps us quickly determine if there is a remainder. A remainder of zero indicates that the number is a root.

• If the remainder is not zero, such as in the first trial with \(x = 1\), this candidate is not a root.
• When tested with \(x = -1\), the remainder becomes zero, indicating it is a valid root.

Once a root is found, synthetic division provides the quotient for further factorization of the polynomial.

After confirming that \(x = -1\) is a root using synthetic division, you are left with a quotient from the polynomial of a lower degree, in this case, \(3x^2 + 8x - 3\). This equation can be solved using the quadratic formula, which is a universal method to find the roots of any quadratic equation—given that the quadratic is in the form \(ax^2 + bx + c = 0\).The quadratic formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]When solving, you plug in the values:

• \(a = 3\)
• \(b = 8\)
• \(c = -3\)

The formula then provides the solutions for \(x\), which, based on our calculations, are \(x = \frac{1}{3}\) and \(x = -3\). This step completes the task of finding the roots that contribute to solving the initial polynomial equation.

The discriminant is an important part of solving a quadratic equation with the quadratic formula. It is found in the expression \(b^2 - 4ac\), under the square root in the quadratic formula. The discriminant determines the nature and number of roots that a quadratic equation has:

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, there is one real root (or a repeated root).
• If the discriminant is negative, there are no real roots, but two complex roots instead.

In the given example, we calculate the discriminant for \(3x^2 + 8x - 3\): \[b^2 - 4ac = 8^2 - 4 \times 3 \times (-3) = 64 + 36 = 100\]This positive discriminant of 100 indicates there are two distinct real roots for the quadratic equation. Using the quadratic formula, these roots can indeed be confirmed as \(x = \frac{1}{3}\) and \(x = -3\), complementing the first identified root, \(x = -1\), of the initial cubic equation.