Problem 22 - Precalculus with Calculus Previews

Question

Use division to show that the indicated polynomial is a factor of the given polynomial function \(f .\) Find all other zeros and then give the complete factorization of \(f(x)\). $$ (x+3)^{2} ; f(x)=x^{4}-4 x^{3}-22 x^{2}+84 x+261 $$

Step-by-Step Solution

Answer

The complete factorization is \((x + 3)^2(x + 1)(x^2 - 8x + 1)\).

Step 1: Set Up the Division

We need to divide the polynomial \( f(x) = x^4 - 4x^3 - 22x^2 + 84x + 261 \) by the factor \( (x + 3) \). We'll perform polynomial long division to check if \( (x+3) \) is indeed a factor.

Step 2: Perform the Division

Start dividing \( x^4 \) by \( x \), which gives \( x^3 \). Multiply \( x^3 \) by \( (x + 3) \) to get \( x^4 + 3x^3 \). Subtract this from the original polynomial to get the new polynomial: \(-7x^3 - 22x^2 + 84x + 261\). Repeat the process until you get a remainder of zero.

Step 3: Complete Polynomial Long Division

Continuing the division, divide \( -7x^3 \) by \( x \) to get \( -7x^2 \). Multiply and subtract as before. The new expression is \(-1x^2 + 84x + 261\). Repeat this process through \(-x + 21\) and finally \(-1\) yielding the complete division: \((x+3)(x^3 - 7x^2 - x - 1) \) with no remainder.

Step 4: Further Factoring the Quotient

Now that we've confirmed \((x + 3)\) is a factor, we have \(f(x) = (x + 3)(x + 3)(x^3 - 7x^2 - x - 1)\). We need to further factor \(x^3 - 7x^2 - x - 1\) if possible. By trying possible rational roots, we find \(x = -1\) is a root.

Step 5: Factor by Synthetic Division

Perform synthetic division of \(x^3 - 7x^2 - x - 1\) by \(x + 1\) to find further factors. This yields \(x^3 - 7x^2 - x - 1 = (x + 1)(x^2 - 8x + 1)\).

Step 6: Verifying and Writing Complete Factorization

Now we have the full factorization: \(f(x) = (x + 3)^2(x + 1)(x^2 - 8x + 1)\). No further factoring of the quadratic term is possible over real numbers.

Key Concepts

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into simpler polynomials that, when multiplied together, give you the original polynomial. This process is crucial for solving polynomial equations, understanding their structure, and finding their roots.
In the given problem, we are tasked with factoring function \( f(x) = x^4 - 4x^3 - 22x^2 + 84x + 261 \), starting by confirming \((x+3)^2\) as a factor.
Factorization often requires using a combination of techniques, such as finding common factors, utilizing synthetic division, or employing special factorization formulas for cases like perfect square trinomials and difference of squares.

This eventually leads to a product of factors which here is \((x + 3)^2(x + 1)(x^2 - 8x + 1)\), where it no longer completely factors over the real numbers, showing the depth of how complex polynomial structures are broken down.

Zeros of Polynomials

The zeros of a polynomial are the values of \(x\) that make the polynomial equal zero. These are also known as 'roots' or 'solutions' and they play a key role in understanding the graph and behavior of polynomial functions.
From the given factorization of the polynomial \( f(x) = (x + 3)^2(x + 1)(x^2 - 8x + 1) \), we can identify zeros as the values where each factor is equal to zero.

• The first factor \((x + 3)^2\) implies that \(x = -3\) is a zero with multiplicity 2.
• The second factor \((x + 1)\) reveals another zero, \(x = -1\).
• For the third factor \(x^2 - 8x + 1 = 0\), you'll typically use the quadratic formula to find the zeros, as it does not factor further neatly over real numbers.

These zeros help in sketching the graph and understanding where the polynomial crosses the x-axis.

Synthetic Division

Synthetic division is a simplified form of polynomial division, particularly useful when dividing by linear factors. It's much quicker than long division and primarily used when verifying possible roots using rational root theorem or finding factors.
In our task, we utilize synthetic division to further decompose the polynomial \(x^3 - 7x^2 - x - 1\) by a factor \(x + 1\).
Only coefficients are utilized in synthetic division, which streamlines calculations significantly. By lining up the coefficients and executing row-wise addition or subtraction, it speeds up finding the quotient and also checks for remainder. If the remainder ends up zero, the divisor is indeed a factor. Synthetic division tells us that when \(x = -1\), it gives a factor of \((x + 1)\), confirming it as a zero too.

Rational Root Theorem

The Rational Root Theorem provides a strategy to identify potential rational roots of a polynomial equation. It states that any potential rational root of the polynomial equation \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0\) is a ratio of the factors of the constant term \(a_0\) over factors of the leading coefficient \(a_n\).

Applying this to \( f(x) = x^4 - 4x^3 - 22x^2 + 84x + 261 \), we determine possible roots by considering the factors of 261 over factors of 1. This method gives a list of potential roots simplifying the process of synthetic division checks.
This theorem is especially powerful for initial testing to simplify polynomials more efficiently by narrowing down the pool of probable zeros. For the task, it helped confirm \(x = -1\) as a zero, streamlining the further factorization into \((x + 1)(x^2 - 8x + 1)\). This approach hones in on possible zeros effectively, reducing unnecessary work.