Problem 42 - Precalculus Mathematics for Calculus

Question

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=6 x^{4}-7 x^{3}-12 x^{2}+3 x+2$$

Step-by-Step Solution

Answer

Rational zeros are \( x = 2, \frac{1}{3} \). Factored form: \((x - 2)(3x - 1)(2x - 1)(3x + 1)\).

Step 1: Identify possible rational zeros

According to the Rational Root Theorem, any rational solution \( \frac{p}{q} \) of the polynomial equation \( P(x) = 0 \) has \( p \) as a factor of the constant term and \( q \) as a factor of the leading coefficient. Here, the constant term is 2, and the leading coefficient is 6. Therefore, the factors of 2 are \( \pm 1, \pm 2 \) and the factors of 6 are \( \pm 1, \pm 2, \pm 3, \pm 6 \). Thus, the possible rational zeros are \( \pm 1, \pm 2, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6} \).

Step 2: Test possible zeros using synthetic division

We test the possible zeros using synthetic division to determine which are actually zeros of the polynomial. After testing, we find that \( x = 2 \) is a zero. Performing synthetic division with \( x = 2 \), the remaining polynomial is \( 6x^3 + 5x^2 - 2x + 1 \).

Step 3: Continue factoring the quotient polynomial

Repeat the process of identifying and testing possible rational zeros for the polynomial \( 6x^3 + 5x^2 - 2x + 1 \). Testing these possible zeros, we find that \( x = \frac{1}{3} \) is a zero. Performing synthetic division with \( x = \frac{1}{3} \), the remaining polynomial is \( 18x^2 + 3x - 3 \).

Step 4: Factor the quadratic polynomial

For the quadratic polynomial \( 18x^2 + 3x - 3 \), we can factor it further. By seeking two numbers whose product is \( 18 \cdot -3 = -54 \) and whose sum is 3, we factor the polynomial as \(( 6x - 3)(3x + 1) \). Therefore, the complete factored form of the original polynomial is \((x - 2)(3x - 1)(6x - 3)(3x + 1) \).

Step 5: Write the polynomial in fully factored form

Finally, simplify the factors to ensure they are in their simplest forms. For example, \((6x - 3) = 3(2x - 1)\). Thus, the polynomial can be rewritten as \((x - 2)(3x - 1)(2x - 1)(3x + 1)\).

Key Concepts

Synthetic Division

Synthetic division is a simplified form of long division used for polynomials. It's particularly beneficial when you need to determine whether a given number is a root of the polynomial. This method is convenient because it reduces complex calculations into a series of additions and multiplications.

To perform synthetic division, you write down the coefficients of the polynomial. The value you are testing as a root (known as the divisor) is placed outside. Then, you systematically bring down numbers, multiply, add, and repeat. This process will determine if the remainder is zero. If it is, the divisor is indeed a zero of the polynomial.

• Begin with the polynomial: rather than dividing x terms directly, use only the coefficients.
• If you suspect a zero, like in our task where you tested values such as +2 and 1/3, use synthetic division to confirm.

Through this method, you can quickly identify rational zeros and reduce the degree of the polynomial, making further factorization easier.

Factoring Polynomials

Factoring polynomials is the process of expressing a polynomial as the product of its simpler polynomials. This method simplifies equations and can help in solving them, especially after discovering one or more zeros.

The general aim in factoring is to break down a complex polynomial into several linear factors, or simpler polynomials, as demonstrated with our earlier equation. For example, breaking down the polynomial results until we have the expression in its simplest multiplicative components.

• Use identified zeros to simplify the polynomial through division.
• Continue to test possible zeros until further division is impossible or unnecessary, and you're left with quadratic or simple terms to factor.

In our scenario, after discovering a few rational zeros, the polynomial was decomposed into a sequence of simpler binomials that were easily combined.

Rational Zeros

Rational zeros are those that can be expressed as fractions, where the numerator and the denominator are integers that divide the constant term and the leading coefficient respectively. The Rational Root Theorem provides a systematic way to find such zeros.

The Rational Root Theorem states that for a polynomial with integer coefficients, any rational zero will have a numerator that is a factor of the constant term and a denominator that is a factor of the leading coefficient.

To find rational zeros, you'll:

• List potential zeros using the theorem, as we did by listing all combinations of factors for both the constant and leading coefficient of the polynomial.
• Use synthetic division to test which potential zero is indeed a zero of the polynomial.

This process narrows down the list of possible solutions quickly, helping you move towards factoring the polynomial completely.

Polynomial Roots

Finding the polynomial roots is crucial in solving polynomial equations. These roots are values of x that make the polynomial equal to zero. They can be rational, irrational, or complex, but rational roots are especially notable because they can be neatly expressed as fractions.

The process to determine all polynomial roots typically begins with identifying potential rational roots using the Rational Root Theorem. After rational roots are found and confirmed via synthetic division, you continue factoring to discover other roots.

• Use identified rational zeros as foundation stones to unravel the entire polynomial's makeup, focusing on simplifying it down.
• Once the polynomial is broken down to its simplest quadratic or linear forms, further algebraic and numerical methods can find any remaining non-rational roots.

In this exercise, finding the rational zeros helped in breaking down the polynomial into factors, ultimately leading to identifying all roots effectively.