Problem 15 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The rational zeros are \( x = 1 \) and \( x = -2 \). The factored form is \( (x - 1)(x + 2)^2 \).

Step 1: Identify Potential Rational Zeros

Use the Rational Root Theorem, which states that any rational zero of the polynomial \( P(x) = x^3 + 3x^2 - 4 \) is a fraction \( \frac{p}{q} \), where \( p \) is a factor of the constant term (-4) and \( q \) is a factor of the leading coefficient (1). This gives us potential rational zeros: \( \pm 1, \pm 2, \pm 4 \).

Step 2: Test Potential Zeros Using Synthetic Division

Perform synthetic division or substitute the potential zeros into the polynomial to find which values make the polynomial equal to zero. After testing, you find that \( x = 1 \) is a zero because when substituted back into the polynomial, \( P(1) = 1^3 + 3(1)^2 - 4 = 0 \).

Step 3: Factor the Polynomial

Using the zero found, \( x = 1 \), factor \( x - 1 \) from the polynomial \( P(x) \). Using synthetic division, divide the polynomial by \( x - 1 \) to find the quotient: \( x^2 + 4x + 4 \). Hence, \( P(x) = (x - 1)(x^2 + 4x + 4) \).

Step 4: Factor the Quadratic Component

Factor the quadratic \( x^2 + 4x + 4 \). Recognize it as a perfect square trinomial: \((x + 2)^2\). So, the polynomial can be rewritten as \( P(x) = (x - 1)(x + 2)^2 \).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a helpful tool to determine possible rational zeros of a polynomial. It states that if a polynomial, like \( P(x) = x^3 + 3x^2 - 4 \), has a rational root \( \frac{p}{q} \), then:

• \( p \) is a factor of the constant term, which in our example is \(-4\).
• \( q \) is a factor of the leading coefficient, which is \(1\) for our polynomial.

Thus, the possible rational roots for this polynomial are all combinations of \( \pm 1, \pm 2, \) and \( \pm 4 \), derived from factoring \(-4\) and \(1\). This theorem not only reduces the number of roots to check but also makes it easier to solve polynomials efficiently.

Synthetic Division

Synthetic division is a simplified form of polynomial division, particularly useful for checking potential roots of polynomials quickly. For our polynomial, after identifying potential zeros from the rational root theorem, we conduct synthetic division:

• Select a potential root, for instance, \( x = 1 \).
• Set up synthetic division using 1 and the polynomial's coefficients \( [1, 3, 0, -4] \).
• Perform synthetic division, and if the remainder is zero, it's a root of the polynomial.

We found \( x = 1 \) to be a root because it results in no remainder, confirming it's a rational zero. Synthetic division also helps to factor the polynomial further.

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be expressed as \( (a + b)^2 \) or \( (a - b)^2 \). It simplifies polynomial expressions and is essential in factoring for simpler solutions.
In our example, after performing synthetic division, we arrived at the quadratic term \( x^2 + 4x + 4 \). Recognizing it as a perfect square trinomial, we notice it matches the pattern:
\((x + 2)^2 = x^2 + 4x + 4\)
This ensures easy factoring and simplifies the polynomial: \( P(x) = (x - 1)(x + 2)^2 \). Spotting perfect square trinomials can significantly reduce the complexity of polynomial expressions.

Polynomial Factoring

Polynomial factoring involves breaking down a polynomial into simpler expressions that can multiply together to reach the original polynomial. Factoring is crucial for solving polynomial equations, as it provides a straightforward method to identify roots.
In this problem, with one root found using the Rational Root Theorem and synthetic division, we wrote the polynomial as \( (x - 1)(x^2 + 4x + 4) \). Recognizing the quadratic as a perfect square trinomial allowed further factoring to \( (x - 1)(x + 2)^2 \).
Thus, the polynomial factoring process turned the polynomial into its simplest form, revealing all its zeros. This transformation helps solve and graph polynomial equations more efficiently.