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To find the real zeros of a polynomial, we can start by applying the Rational Root Theorem. This theorem is a helpful tool for identifying possible rational zeros of a polynomial equation. According to the theorem, if a polynomial has rational zeros, they are of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term, and \( q \) is a factor of the leading coefficient.
For the polynomial \( P(x) = x^4 - 6x^3 + 4x^2 + 15x + 4 \), the constant term is 4 and the leading coefficient is 1.
This simplification means that possible rational zeros are the factors of 4: \( \pm 1, \pm 2, \pm 4 \). These are the numbers we test to see if they are actual zeros of the polynomial.

Synthetic division is a simple and fast method used to divide polynomials. It is especially useful when you have already found a zero of the polynomial. After identifying potential zeros, such as \(-1\) and \(2\) for our polynomial \( P(x) \), we use synthetic division to divide the polynomial and find smaller, more manageable degrees of polynomials.
Let's break this down:

• First, we perform synthetic division using one of the zeros like \( x = 2 \), which gives us a quotient of a lower degree polynomial, \( x^3 - 4x^2 - 4x + 2 \).
• Then, we further divide using the next zero, \( x = -1 \), resulting in an even simpler quadratic polynomial, \( x^2 - 5x + 2 \).

This reduction is crucial because it makes it easier to use other methods, such as solving a quadratic equation, to find the remaining zeros.

After reducing the polynomial to a quadratic form using synthetic division, we often turn to the quadratic formula to find its zeros. The quadratic formula provides solutions for any quadratic equation of the form \( ax^2 + bx + c = 0 \). The formula is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].For the quadratic polynomial \( x^2 - 5x + 2 \), the coefficients are \( a = 1 \), \( b = -5 \), and \( c = 2 \).
Plugging these values into the quadratic formula, we calculate the real roots:

• \( x = \frac{5 \pm \sqrt{17}}{2} \).

These results give us the additional real zeros, completing the solution for the original polynomial.

Real zeros of a polynomial are simply the values of \( x \) for which the polynomial evaluates to zero. Finding all real zeros is essential as it gives us insight into the polynomial's behavior and graph.
From our example polynomial \( P(x) = x^4 - 6x^3 + 4x^2 + 15x + 4 \), by using the Rational Root Theorem, synthetic division, and the quadratic formula, we determined the real zeros as:

• \( x = -1 \)
• \( x = 2 \)
• \( x = \frac{5 + \sqrt{17}}{2} \)
• \( x = \frac{5 - \sqrt{17}}{2} \)

Identifying real zeros is crucial for understanding where the polynomial crosses or touches the x-axis on a graph.