Problem 18 - Algebra and Trigonometry

Question

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$ P(x)=x^{3}+4 x^{2}-3 x-18 $$

Step-by-Step Solution

Answer

The rational zeros are 2 and -3. Factored form: \( (x - 2)(x + 3)^2 \).

Step 1: Identify Possible Rational Zeros

For polynomial \( P(x) = x^3 + 4x^2 - 3x - 18 \), use the Rational Root Theorem. The potential rational zeros are the factors of the constant term (-18) divided by the factors of the leading coefficient (1). Thus, possible zeros are \( \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18 \).

Step 2: Test Possible Zeros

Test these possible rational zeros by evaluating \( P(x) \) at each of these values.- \( P(1) = 1^3 + 4(1)^2 - 3(1) - 18 = -16 \)- \( P(-1) = (-1)^3 + 4(-1)^2 - 3(-1) - 18 = -16 \)- \( P(2) = 2^3 + 4(2)^2 - 3(2) - 18 = 0 \)Thus, \( x = 2 \) is a zero.

Step 3: Perform Synthetic Division

Use synthetic division to divide \( P(x) \) by \( x - 2 \), since \( 2 \) is a zero:\[\begin{array}{r|rrrr}2 & 1 & 4 & -3 & -18 \ & & 2 & 12 & 18 \\hline & 1 & 6 & 9 & 0 \\end{array}\]The remainder is 0, and the quotient is \( x^2 + 6x + 9 \).

Step 4: Factor the Quotient

The quotient \( x^2 + 6x + 9 \) can be factored by recognizing it as a perfect square:\( x^2 + 6x + 9 = (x+3)^2 \).

Step 5: Write the Polynomial in Factored Form

Combine the factors from the previous steps. The polynomial in factored form is:\( P(x) = (x - 2)(x + 3)^2 \).

Step 6: List All Rational Zeros

The rational zeros of the polynomial are those values of \( x \) that make each factor equal zero. From \( (x - 2)(x + 3)^2 \), the zeros are:- \( x = 2 \) - \( x = -3 \)

Key Concepts

Synthetic Division

Synthetic division is a streamlined method of dividing polynomials, especially handy when you need to divide by a linear factor like \( x - c \). It reduces the complexity of long division by focusing only on the coefficients. In our example, we identified \( x = 2 \) as a root of the polynomial using the Rational Root Theorem. With synthetic division, we used this root to divide the polynomial \( P(x) = x^3 + 4x^2 - 3x - 18 \).

• First, we align the coefficients of the polynomial. For \( P(x) \), the coefficients are 1, 4, -3, and -18.
• Then, we perform the synthetic division using 2 as the divisor. This involves a process of multiplication and addition to find the quotient's coefficients.
• Finally, synthetic division gives a remainder. If this remainder is zero, \( x = 2 \) is a root, confirming our earlier finding.

The result from the division is a quadratic polynomial, \( x^2 + 6x + 9 \), which simplifies solving by factoring to completely understand the roots of the original polynomial.

Polynomial Factoring

Polynomial factoring is the process of breaking down a polynomial into simpler polynomials whose product equals the original polynomial. It's a crucial step in finding zeros of the polynomial. Once we performed synthetic division, we obtained the quadratic polynomial \( x^2 + 6x + 9 \).

• Recognizing this quadratic as a perfect square trinomial is key. Here, the expression can be rewritten as \((x + 3)^2\).
• Factoring allows us to express the original cubic polynomial \( P(x) \) as \((x - 2)(x + 3)^2\).

This factored form is not only simpler but also reveals more about the polynomial's behavior, especially helpful for graphing or solving inequalities.

Rational Root Theorem

The Rational Root Theorem is an essential tool for finding potential rational zeros of a polynomial. It states that any rational root, represented as \( \frac{p}{q} \), must have \( p \) as a factor of the constant term and \( q \) as a factor of the leading coefficient.

• In the polynomial \( P(x) = x^3 + 4x^2 - 3x - 18 \), the constant term is \(-18\) and the leading coefficient is 1.
• This gives possible rational zeros of \( \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18 \).

By testing these possibilities, we found that \( x = 2 \) is a zero. This theorem is a powerful method to shortlist and test possible zeros before more intensive techniques like synthetic division.

Perfect Square Trinomials

Perfect square trinomials are a specific kind of quadratic expression that can be expressed as the square of a binomial. These trinomials take the form \( a^2 + 2ab + b^2 = (a + b)^2 \).
In our example, after using synthetic division, the resulting quadratic factor was \( x^2 + 6x + 9 \). This can be identified as a perfect square trinomial:

• The expression can be rewritten as \((x + 3)^2\), where \(a = x\) and \(b = 3\).
• Recognizing this structure simplifies the factoring process significantly, allowing quicker solutions.

Recognizing and applying the formula for perfect square trinomials helps in swiftly transforming complex polynomials into more manageable forms for further analysis or solving.