Problem 25 - Calculus Single Variable

Question

Write the given polynomial as a product of irreducible polynomials of degree one or two. \(x^{4}+2 x^{3}-2 x^{2}-8 x-8\)

Step-by-Step Solution

Answer

The polynomial is factored as \((x + 1)(x - 2)(x^2 + 3)\).

Step 1: Check for Rational Roots

Use the Rational Root Theorem to find possible rational roots of the polynomial. The Rational Root Theorem suggests that if a polynomial has a rational root, it will be a factor of the constant term divided by the leading coefficient. Here, the constant term is -8 and the leading coefficient is 1, so the possible rational roots are ±1, ±2, ±4, ±8.

Step 2: Test Possible Roots by Substitution

Substitute each possible rational root into the polynomial to see if it equals zero. By trying these, we find that \(x = -1\) and \(x = 2\) are roots because substituting them into the polynomial yields zero.

Step 3: Perform Synthetic Division

Divide the polynomial by \(x + 1\) and \(x - 2\) using synthetic division to factor the polynomial. Dividing by \(x + 1\) first, we get \(x^3 + x^2 - 3x - 8\). Dividing this result by \(x - 2\), we are left with \(x^2 + 3\).

Step 4: Factor Completely

The polynomial has been factored as \((x + 1)(x - 2)(x^2 + 3)\). We check for irreducibility: \(x^2 + 3\) is irreducible over the reals (since it has no real roots), so the factorization is complete.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a useful tool to find potential rational roots of a polynomial. This theorem gives us a way to narrow down which numbers might be roots. To apply it, list all the factors of the constant term and the leading coefficient.
In the exercise, the polynomial is given as \(x^{4} + 2x^{3} - 2x^{2} - 8x - 8\). The constant term is -8, and the leading coefficient is 1. So, the possible rational roots are ±1, ±2, ±4, and ±8.
Once you have this list, substitute these values into the polynomial to check if they are actual roots. A root is a number that, when substituted for \(x\), makes the polynomial equal to zero.

Synthetic Division

Synthetic division is a straightforward method to divide polynomials once you've figured out a root using something like the Rational Root Theorem. It simplifies the process compared to long division.
To perform synthetic division, write down the coefficients of the polynomial. Then use the root of interest, let's say \(k\), as the divisor.

• Drop the first coefficient down to begin the process.
• Multiply this result by \(k\), and add it to the next coefficient.
• Repeat this process until all coefficients have been used.

This helps in breaking down the polynomial in a step-by-step manner, resulting in a new polynomial of lesser degree with a remainder of zero if \(k\) is a true root.
In the example, we divided the original polynomial \(x^{4} + 2x^{3} - 2x^{2} - 8x - 8\) by \(x + 1\) and then by \(x - 2\), leading to \(x^2 + 3\).

Irreducible Polynomials

An irreducible polynomial is one that cannot be factored further into polynomials of lower degree with rational coefficients. For instance, \(x^2 + 3\) can't be broken down any further because it has no real roots: any attempts would involve complex numbers.
When factoring polynomials, especially over the real numbers, it's crucial to identify when a factor can't be simplified further. This ensures the factorization is complete.
In our exercise, after performing synthetic division, we are left with \(x^2 + 3\) as a factor. Since this quadratic doesn't have real roots, it is irreducible, which confirms the completion of our factorization.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable \(x\) in the polynomial expression. It's a fundamental property of the polynomial that dictates its behavior and the number of roots it can have.
Knowing the degree helps predict the possible number of roots and, therefore, the potential complexity of factorizations.
In the provided exercise, the original polynomial has a degree of 4, indicating it could have up to four roots. By identifying these roots and using methods like synthetic division, we can break down the polynomial by finding and using factors until we reach a fully factored form.
Ultimately, the degree provides a framework for understanding the overall shape and possible solutions of the polynomial when graphed.