Problem 11 - Algebra and Trigonometry

Question

Find all rational zeros of the polynomial. $$ P(x)=x^{3}+3 x^{2}-4 $$

Step-by-Step Solution

Answer

The rational zeros are \( x = 1 \) and \( x = -2 \).

Step 1: Identify the Potential Rational Zeros

According to the Rational Root Theorem, any rational zero of the polynomial \( P(x) = x^3 + 3x^2 - 4 \) must be a factor of the constant term divided by one (since the leading coefficient is 1). Therefore, potential rational zeros are the factors of \(-4\), which are \( \pm 1, \pm 2, \pm 4 \).

Step 2: Test Potential Zeros Using Synthetic Division

First, test \( x = 1 \). Perform synthetic division with \( 1 \) and the coefficients \( 1, 3, 0, -4 \). The result of the synthetic division shows no remainder, so \( x = 1 \) is a root. Perform the same process with \( x = -1, 2, -2, 4, -4 \). After testing, only \( x = 1 \) results in a remainder of zero.

Step 3: Factor the Polynomial

Since \(x = 1\) is a root, we can factor \( x - 1 \) out of the polynomial. Using synthetic division with \( x = 1 \), we divide \( P(x) \) to obtain the quotient \( x^2 + 4x + 4 \). Thus, \( P(x) = (x - 1)(x^2 + 4x + 4) \).

Step 4: Factorize the Quadratic

Now, factor \( x^2 + 4x + 4 \). Notice it is a perfect square trinomial that can be written as \( (x + 2)^2 \). Therefore, \( P(x) = (x - 1)(x + 2)^2 \).

Step 5: Write All Rational Zeros

The rational zeros of \( P(x) \) are the solutions to \( (x - 1)(x + 2)^2 = 0 \). These are \( x = 1 \) and \( x = -2 \).

Key Concepts

Polynomial Division

Polynomial division is a process similar to long division, but it involves polynomials instead of numbers. The main goal is to divide a polynomial by another polynomial, often of a lower degree. This is particularly useful for simplifying expressions and finding factors. In the context of the original exercise, we used polynomial division to check potential zeros of the polynomial.

• Set up the division by arranging terms in descending order of power.
• Divide the leading term of the dividend by the leading term of the divisor.
• This result becomes the first term of the quotient.
• Multiply back the entire divisor by this term and subtract from the original polynomial.
• Repeat the process with the new polynomial formed.

Polynomial division helps in breaking down complex polynomials into simpler parts, making it easier to identify roots.

Synthetic Division

Synthetic division is a streamlined, quicker method used to divide a polynomial by a binomial of the form \( (x - c) \). It's especially handy when using the Rational Zero Theorem to test potential rational zeros.
Here's how to perform synthetic division:

• Write down the coefficients of the polynomial.
• Use \( c \), the value you are testing as a potential zero, to perform the division.
• Below the coefficients, bring down the first coefficient unchanged.
• Multiply this value by \( c \) and add it to the next coefficient.
• Repeat the process for all coefficients. If the last value calculated is zero, then \( c \) is a root.

In the exercise, synthetic division helped determine that \( x = 1 \) was a rational zero without much computation complexity.

Polynomial Factoring

Polynomial factoring involves rewriting a polynomial as a product of its factors. This process is crucial for simplifying polynomial expressions and finding solutions to polynomial equations.
The goal in factoring is to express the polynomial in a form where each factor is a polynomial of lower or equal degree. In the exercise, once \( x = 1 \) was identified as a root, the polynomial \( P(x) = x^3 + 3x^2 - 4 \) was factored as \( (x - 1)(x^2 + 4x + 4) \).

• Identify common factors in the polynomial terms.
• Use algebraic identities or factoring techniques to simplify.
• Apply perfect square or difference of squares where applicable.

Factoring allows us to simplify the problem, making it easier to solve for the roots of the polynomial.

Perfect Square Trinomial

A perfect square trinomial is a special form of polynomial that can be expressed as the square of a binomial. It follows the pattern \( a^2 \pm 2ab + b^2 = (a \pm b)^2 \).
In the provided problem, after dividing the original polynomial by \( x - 1 \), the resulting quadratic \( x^2 + 4x + 4 \) is a perfect square trinomial. It factors into \((x + 2)^2\).
Characteristics of perfect square trinomials include:

• The first and last terms are perfect squares.
• The middle term is twice the product of the square roots of the first and last terms.

Recognizing perfect square trinomials is useful in simplifying polynomial expressions and finding solutions, as it reduces the problem to solving simpler equations.