Problem 26 - Precalculus Mathematics for Calculus

Question

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=x^{4}-2 x^{3}-3 x^{2}+8 x-4$$

Step-by-Step Solution

Answer

The rational zeros are 1, 2, and -2. The polynomial factors as \((x - 1)^2(x - 2)(x + 2)\).

Step 1: Identify Possible Rational Zeros

To find all rational zeros, use the Rational Root Theorem. According to this theorem, any potential rational root \( \frac{p}{q} \) must satisfy that \( p \) is a factor of the constant term (-4) and \( q \) is a factor of the leading coefficient (1). The potential rational zeros are thus \( \pm 1, \pm 2, \pm 4 \).

Step 2: Test Possible Zeros

Substitute each potential zero into the polynomial \( P(x) = x^4 - 2x^3 - 3x^2 + 8x - 4 \):- \( P(1) = 1^4 - 2 \cdot 1^3 - 3 \cdot 1^2 + 8 \cdot 1 - 4 = 0 \)- \( P(-1) = (-1)^4 - 2 \cdot (-1)^3 - 3 \cdot (-1)^2 + 8 \cdot (-1) - 4 eq 0 \)- \( P(2) = 2^4 - 2 \cdot 2^3 - 3 \cdot 2^2 + 8 \cdot 2 - 4 = 0 \)- \( P(-2) = (-2)^4 - 2 \cdot (-2)^3 - 3 \cdot (-2)^2 + 8 \cdot (-2) - 4 = 0 \)Thus, 1, 2, and -2 are zeros.

Step 3: Divide the Polynomial by (x - 1)

Use synthetic division to divide \( P(x) = x^4 - 2x^3 - 3x^2 + 8x - 4 \) by \( x - 1 \), since \( x = 1 \) is a zero. This results in a quotient of \( x^3 - x^2 - 4x + 4 \).

Step 4: Divide the Quotient by (x - 2)

Apply synthetic division again to divide \( x^3 - x^2 - 4x + 4 \) by \( x - 2 \). This results in a second quotient \( x^2 + x - 2 \).

Step 5: Factor the Quadratic

Factor \( x^2 + x - 2 \). The factorized form is \((x + 2)(x - 1)\).

Step 6: Write the Polynomial in Factored Form

Combine all factors: since \( 1, 2, \) and \(-2\) are roots, and the quadratic factor is \((x + 2)(x - 1)\), the polynomial can be written as \((x - 1)^2(x - 2)(x + 2)\).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a handy tool for predicting possible rational roots of a polynomial with integer coefficients. Here’s how it works:

• The theorem states that any possible rational root of a polynomial equation of the form \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 = 0 \) is of the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers.
• \( p \) represents a factor of the constant term \( a_0 \), while \( q \) is a factor of the leading coefficient \( a_n \).

In the given exercise, our polynomial is \( P(x) = x^4 - 2x^3 - 3x^2 + 8x - 4 \). The constant term here is -4 and the leading coefficient is 1.
Hence the potential rational zeros are factors \( \pm 1, \pm 2, \pm 4 \). We test these values to find any actual zeros of the polynomial. This narrows down our options and helps us understand the structure of the polynomial.

Polynomial Factoring

Factoring a polynomial means expressing it as a product of simpler polynomials, which makes it easier to solve or simplify the equation. Factoring involves breaking down the polynomial equation into its basic building blocks.

• The zeros of the polynomial represent the x-values where the polynomial equals zero.
• Once the zeros are identified, they can be used to construct factors of the polynomial.

In the given problem, after using the Rational Root Theorem, the zeros \( 1, 2, \) and \(-2\) are found. By factoring the polynomial based on these zeros, we can express it as products of linear factors:
\((x - 1), (x - 2),\) and \((x + 2)\).
For the remainder polynomial \(x^2 + x - 2\), which results from synthetic division, we further factor it into \((x + 2)(x - 1)\). Combine all these factors correctly to get the full factored form \((x - 1)^2(x - 2)(x + 2)\). This process of polynomial factoring simplifies the polynomial into components that are easy to utilize for further analysis or solutions.

Synthetic Division

Synthetic division is a simplified way of dividing a polynomial by a linear divisor of the form \(x - c\), where \(c\) is a constant. It's especially useful after determining potential zeros using the Rational Root Theorem.

• Instead of dividing the polynomial directly, synthetic division uses coefficients and involves less writing.
• It quickly reveals any remainder, which helps confirm if a number is a root of the polynomial.

To solve the exercise problem, we performed synthetic division with \(x - 1\) since \(1\) is a root. We reduced \(x^4 - 2x^3 - 3x^2 + 8x - 4\) to a simpler polynomial \(x^3 - x^2 - 4x + 4\).
This simpler polynomial was further divided by \(x - 2\), giving \(x^2 + x - 2\). Using synthetic division helps break down the polynomial efficiently into more manageable parts, confirming zeros and paving the way for easier factoring.