Problem 48 - Algebra and Trigonometry

Question

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(\mathrm{a}) .\) $$ P(x)=x^{3}-5 x^{2}+2 x+12 $$

Step-by-Step Solution

Answer

The real zeros are \(x = 3\), \(x = 1 + \sqrt{5}\), and \(x = 1 - \sqrt{5}\).

Step 1: Identify Possible Rational Zeros

According to the Rational Root Theorem, the possible rational zeros of the polynomial \(P(x) = x^3 - 5x^2 + 2x + 12\) are the factors of the constant term (12) divided by the factors of the leading coefficient (1). This gives the possible rational zeros as \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \).

Step 2: Use Synthetic Division to Test Possible Zeros

Use synthetic division to determine if any of the possible rational zeros are actual zeros of the polynomial. Start testing from +1, -1, +2, -2, etc. By trial with +3, the resulting synthetic division yields a remainder of 0, confirming that \(x = 3\) is a zero.

Step 3: Factor the Polynomial

With \(x = 3\) confirmed as a zero, factor the polynomial as \((x - 3) \times (x^2 - 2x - 4)\). This results from dividing the original polynomial by \(x - 3\), finding that the quotient is \(x^2 - 2x - 4\).

Step 4: Find Zeros of the Quadratic Factor

Solve \(x^2 - 2x - 4 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here \(a = 1\), \(b = -2\), \(c = -4\). Calculate the discriminant \((b^2 - 4ac)\) as \(4 + 16 = 20\). The zeros are \(x = \frac{2 \pm \sqrt{20}}{2} = 1 \pm \sqrt{5}\).

Step 5: List All Real Zeros

The real zeros of the polynomial are \(x = 3\), \(x = 1 + \sqrt{5}\), and \(x = 1 - \sqrt{5}\).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a very helpful tool when finding the real zeros of a polynomial. It provides a way to list all possible rational zeros based on the coefficients of the polynomial. For a polynomial of the form \(P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\):

• The possible rational zeros are the factors of the constant term \(a_0\) divided by the factors of the leading coefficient \(a_n\).

For the given polynomial \(P(x) = x^3 - 5x^2 + 2x + 12\), this means our constant term is 12 and the leading coefficient is 1.
By applying the Rational Root Theorem, we determine the possible rational zeros as \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \).
This approach narrows down the number of values to check, making solving much less daunting.

synthetic division

Synthetic division is a quick method for dividing a polynomial by a binomial of the form \(x - c\). It is particularly useful when testing the possible rational zeros. Here's how it works:

• Write down the coefficients of the polynomial.
• Bring down the lead coefficient to a new line.
• Multiply by \(c\) (the value being tested) and add to the next coefficient, repeating until completion.

When testing \(x = 3\) as a possible zero for \(P(x) = x^3 - 5x^2 + 2x + 12\), we found that the remainder was zero using synthetic division.
This verification step confirmed that \(x = 3\) is indeed a zero of the polynomial. It's a useful shortcut that saves time compared to polynomial long division.

quadratic formula

The quadratic formula is essential for solving quadratic equations that cannot be easily factored. When the polynomial has been reduced to a quadratic form or part, this formula helps find the exact zeros.The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]For the given quadratic \(x^2 - 2x - 4 = 0\), plug in the coefficients \(a = 1\), \(b = -2\), and \(c = -4\).

• Calculate the discriminant: \(b^2 - 4ac\), which in this case is 20.
• Apply the formula to get the solutions \(x = \frac{2 \pm \sqrt{20}}{2}\).

This formula is a powerful tool because it gives us both real and complex solutions, providing a complete picture of the possible solutions.
For this exercise, it helped find the real zeros \(x = 1 + \sqrt{5}\) and \(x = 1 - \sqrt{5}\).

factoring polynomials

Factoring polynomials is the process of breaking down a complex expression into simpler components, often revealing the polynomial's zeros. When we identify one zero, like \(x = 3\), we can factor the polynomial by using division techniques (such as synthetic division) to simplify the expression.In the case of \(P(x) = x^3 - 5x^2 + 2x + 12\), knowing \(x = 3\) is a zero allowed us to divide the polynomial by \(x - 3\), simplifying it to \((x - 3)(x^2 - 2x - 4)\).

• From there, the quadratic factor \(x^2 - 2x - 4\) can be further solved to find more zeros.
• This demonstrates the interconnectedness of factoring, synthetic division, and using the quadratic formula.

The goal is to get the polynomial into a form where each component is easy to solve, enabling us to find all the real zeros efficiently.