Problem 45 - Precalculus Mathematics for Calculus

Question

Find all zeros of the polynomial. $$P(x)=x^{3}+2 x^{2}+4 x+8$$

Step-by-Step Solution

Answer

The zeros are \(-2, 2i,\) and \(-2i\).

Step 1: Understanding the Problem

We need to find all zeros of the polynomial \( P(x) = x^3 + 2x^2 + 4x + 8 \). Zeros are the values of \( x \) that make \( P(x) = 0 \).

Step 2: Finding Rational Zeros

Use the Rational Root Theorem to find possible rational zeros. The theorem suggests that the possible rational zeros are the factors of the constant term (8) divided by factors of the leading coefficient (1). Therefore, the possible rational zeros are \( \pm 1, \pm 2, \pm 4, \pm 8 \).

Step 3: Testing Possible Zeros

Test each possible rational zero using substitution or synthetic division. If we substitute \( x = -2 \) into \( P(x) \), we get: \( P(-2) = (-2)^3 + 2(-2)^2 + 4(-2) + 8 = -8 + 8 - 8 + 8 = 0 \). Thus, \( x = -2 \) is a zero.

Step 4: Factor the Polynomial

Since \( x = -2 \) is a zero, \( x + 2 \) is a factor of \( P(x) \). To find the other factors, perform synthetic division of \( P(x) \) by \( x + 2 \). The result is \( x^2 + 4 \), so \( P(x) = (x + 2)(x^2 + 4) \).

Step 5: Solve the Quadratic

Solve \( x^2 + 4 = 0 \) for \( x \). Rearranging gives \( x^2 = -4 \). Taking the square root of both sides, we find \( x = \pm 2i \). Thus, the quadratic has complex roots.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a handy tool when trying to find the zeros of a polynomial. It helps you identify potential rational zeros quickly without having to test every possible number. The theorem states that any rational solution, in the simplest form of \( \frac{p}{q} \), must have a numerator, \( p \), that is a factor of the constant term and a denominator, \( q \), that is a factor of the leading coefficient.

• In our exercise, the polynomial is \( P(x) = x^3 + 2x^2 + 4x + 8 \).
• The constant term is 8, and the leading coefficient is 1.
• So, the possible rational zeros are the factors of 8 divided by the factors of 1.

Therefore, the possible rational zeros are \( \pm 1, \pm 2, \pm 4, \pm 8 \). This means that if there are any rational roots, they will be among these values.

Synthetic Division

Synthetic Division is a simplified form of polynomial division, particularly useful for dividing polynomials by linear factors. This method makes the process quicker and requires less computation than long division. It's especially helpful when testing the possible zeros given by the Rational Root Theorem. Let's dive into how it works with our polynomial.

• First, choose a potential zero to test. From our list, let's try \( x = -2 \).
• Set up the synthetic division by writing down the coefficients of the polynomial: 1, 2, 4, and 8.
• Start by bringing down the leading coefficient (1).
• Multiply it by \( -2 \) and add the result to the next coefficient.

Continue this process down the row. If \( x = -2 \) is indeed a root, you will get a remainder of 0. In this case, it works out perfectly, indicating that \( x + 2 \) is a factor of the polynomial.

Complex Roots

Complex Roots arise in polynomial equations when you have to solve something like a quadratic with no real solutions. These roots appear when you have a negative number under a square root. In our exercise, after finding that \( x + 2 \) is a root, we divided the polynomial and got \( x^2 + 4 \).

• To solve \( x^2 + 4 = 0 \), first rearrange it to \( x^2 = -4 \).
• Finding the square root of both sides gives \( x = \pm \sqrt{-4} = \pm 2i \).

Here, \( i \) is the imaginary unit, defined by \( i^2 = -1 \). Thus, our complex roots are \( 2i \) and \( -2i \). These are always found in pairs, and when a polynomial has real coefficients, non-real roots come in conjugate pairs.

Polynomial Factorization

Polynomial Factorization involves expressing a polynomial as a product of its factors. This is a powerful technique that not only helps find the roots but also simplifies calculations in many algebraic problems. For \( P(x) = x^3 + 2x^2 + 4x + 8 \), the factorization process is as follows:

• We have already determined that \( x + 2 \) is a factor, thanks to \( x = -2 \) being a root.
• Using synthetic division, we obtained the quotient \( x^2 + 4 \) after dividing by \( x + 2 \).
• Thus, the polynomial can be rewritten as \( P(x) = (x + 2)(x^2 + 4) \).
• The factor \( x^2 + 4 \) can be further factorized in terms of complex roots as \( x^2 + (2i)^2 \).

This rewriting helps us see the relationship between the polynomial's factors and its zeros, both real and complex, enhancing our understanding and ability to manipulate polynomial expressions.