Problem 43 - Algebra and Trigonometry

Question

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example 3(a). $$ P(x)=x^{4}-6 x^{3}+4 x^{2}+15 x+4 $$

Step-by-Step Solution

Answer

The real zeros are \(x = -1, x = 2, x = \frac{5 \pm \sqrt{17}}{2}\).

Step 1: Check for Possible Rational Zeros

According to the Rational Root Theorem, any rational zero of the polynomial \(P(x) = x^4 - 6x^3 + 4x^2 + 15x + 4\) must be a factor of the constant term (4) divided by a factor of the leading coefficient (1). Thus, possible rational zeros are \(\pm 1, \pm 2, \pm 4\).

Step 2: Test Possible Rational Zeros

Substitute the possible rational zeros into the polynomial:- \(P(1) = 1^4 - 6\cdot1^3 + 4\cdot1^2 + 15\cdot1 + 4 = 18\) (not zero)- \(P(-1) = (-1)^4 - 6(-1)^3 + 4(-1)^2 + 15(-1) + 4 = 0\)- \(P(2) = 2^4 - 6\cdot2^3 + 4\cdot2^2 + 15\cdot2 + 4 = 0\)- \(P(-2)\) and \(P(4)\) are calculated and found to be non-zero.Thus, \(x = -1\) and \(x = 2\) are real zeros.

Step 3: Use Polynomial Division

Since \(x = 2\) and \(x = -1\) are zeros, we can use synthetic division to divide the polynomial by \(x+1\) and \(x-2\):First, divide by \(x - 2\):The quotient is \(x^3 - 4x^2 - 4x + 2\).Next, divide the result by \(x + 1\):The quotient is \(x^2 - 5x + 2\).

Step 4: Solve the Quadratic Equation

We need to solve the quadratic equation \(x^2 - 5x + 2 = 0\). Using the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \(a = 1\), \(b = -5\), \(c = 2\).Plugging in these values:\[ x = \frac{5 \pm \sqrt{25 - 8}}{2} = \frac{5 \pm \sqrt{17}}{2} \]These are the remaining two real zeros.

Key Concepts

Rational Root Theorem

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

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.

quadratic formula

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

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.