Problem 20 - Precalculus Mathematics for Calculus

Question

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=x^{3}-x^{2}-8 x+12$$

Step-by-Step Solution

Answer

The rational zeros of the polynomial are 2 and -3; factored form is \((x-2)(x+3)^2\).

Step 1: Understanding the Polynomial

We need to find the rational zeros of the polynomial \( P(x) = x^3 - x^2 - 8x + 12 \). A rational zero is a zero that can be expressed as a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers.

Step 2: Applying the Rational Root Theorem

According to the Rational Root Theorem, any rational zero of \( P(x) \) is of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term (12) and \( q \) is a factor of the leading coefficient (1). Since \( q \) must be \( \pm 1 \), the possible rational roots are factors of 12: \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \).

Step 3: Testing Potential Rational Zeros

We need to test each possible rational zero using synthetic division or by direct substitution into the polynomial to see if it yields zero. Testing \( x = 1 \): \[ P(1) = 1^3 - 1^2 - 8 \times 1 + 12 = 4 \]. \( x = 1 \) is not a zero. Testing \( x = 2 \): \[ P(2) = 2^3 - 2^2 - 8 \times 2 + 12 = 0 \]. \( x = 2 \) is a zero. Testing \( x = -3 \): \[ P(-3) = (-3)^3 - (-3)^2 - 8 \times (-3) + 12 = 0 \]. \( x = -3 \) is a zero. Testing other values won't return further rational zeros.

Step 4: Finding the Quotient Polynomial

Since we found \( x = 2 \) and \( x = -3 \) are zeros, we should perform polynomial division of \( P(x) \) by \( (x-2)(x+3) = x^2 + x - 6 \) to find the quotient polynomial. After division, we find the quotient is \( x - 2 \).

Step 5: Writing the Polynomial in Factored Form

Since \( x = 2 \) and \( x = -3 \) were zeros, and the quotient is \( x-2 \), \( P(x) \) can be expressed in factored form as \( (x-2)(x+3)^2 \).

Step 6: Verifying the Solution

Multiply out \( (x-2)(x+3)^2 \) to verify it returns the original polynomial. \[ (x^2 + 6x + 9)(x - 2) = x^3 - x^2 - 8x + 12 \]. This confirms the factorization is correct.

Key Concepts

Polynomial Division

Polynomial division is a technique used to divide one polynomial by another, similar to how we divide numbers. Just as with numbers, the division of polynomials yields a quotient and possibly a remainder. In our exercise, after determining some zeros, the remaining factor of the polynomial needs to be found using division. The goal is to simplify the polynomial into a product of smaller polynomials, which represent its factored form.
For example, in the exercise, once we identified that 2 and -3 are zeros of the polynomial, the polynomial can be divided by y the factor \((x-2)(x+3)\). This is done to determine the quotient, which tells us what remains of the polynomial after considering the known zeros. Polynomial division helps in finding this quotient, which is essential for completely factoring the polynomial.
There are different methods of polynomial division, including long division, which is quite similar to regular division of numbers, and synthetic division, which is a shortcut method useful for dividing by linear factors.

Factored Form

The factored form of a polynomial expresses the polynomial as a product of its factors. It provides a clear view of the polynomial's zeros and helps in solving equations and graphing. When a polynomial is in factored form, it is written as \[ (x-a_1)(x-a_2)...(x-a_n) \] where each \(a_i\) is a root (zero) of the polynomial.
In the given problem, after using the Rational Root Theorem and synthetic division, the polynomial \(P(x) = x^3 - x^2 - 8x + 12\) was rewritten in the form \((x-2)(x+3)^2\). This form makes it clear that the polynomial has zeros at \(x = 2\) and \(x = -3\), with \(x = -3\) being a repeated root, as indicated by the square. Factored form is useful not only for solving equations but also for quickly identifying any number of important characteristics of the polynomial, such as end behavior or intercepts.
Factored form is a powerful tool in algebra because it highlights the zeros that are crucial for understanding the behavior and nature of a polynomial.

Synthetic Division

Synthetic division is a simplified form of polynomial division, specifically designed to divide polynomials by linear divisors of the form \(x - c\). It is a fast and efficient method that uses less writing and calculation than traditional long division.
To divide using synthetic division, follow these steps:

• Write down the coefficients of the polynomial in descending order of power.
• Select the zero of the divisor (for \(x - c\), it is \(c\)).
• Use this value to sequentially multiply and add within the coefficients, effectively reducing the polynomial by one degree.

For instance, in finding the zeros of \(P(x)=x^3-x^2-8x+12\), synthetic division confirms \(x = 2\) as a root by showing that the remainder is zero. This technique allows us to test possible rational zeros found via the Rational Root Theorem efficiently.
Synthetic division is especially useful when dealing with polynomials with real number coefficients, speeding up the factorization process while providing a clear path towards zero discovery.