Problem 25 - Algebra and Trigonometry

Question

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$ P(x)=x^{4}-5 x^{2}+4 $$

Step-by-Step Solution

Answer

The rational zeros are \( x = 1, -1, 2, -2 \); Factored form is \( (x - 1)(x + 1)(x - 2)(x + 2) \).

Step 1: Identify Potential Rational Zeros

Use the Rational Root Theorem which states that any rational zero of a polynomial \( P(x) \) with integer coefficients is of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient. Here, the constant term is 4 and the leading coefficient is 1. So, the potential rational zeros are \( \pm 1, \pm 2, \pm 4 \).

Step 2: Test Potential Zeros Using Synthetic Division

Test each of the potential rational zeros by evaluating the polynomial or using synthetic division. Start with \( x = 1 \). Synthetic division of \( P(x) \, \) using \( x = 1 \) as a potential zero, results in a remainder of 0, meaning \( x = 1 \) is one root. Repeat for other values \( x = -1, 2, -2, 4, -4 \). Use synthetic division until all real zeros are found.

Step 3: Determine All Real Zeros

Through synthetic division, the rational zeros obtained are \( x = 1 \) and \( x = -1 \). The polynomial \( P(x) \) can be written as \( (x - 1)(x + 1)(x^2 - 4) \), where \( x^2 - 4 \) is further factored as \( (x - 2)(x + 2) \). Hence, the zeros are \( x = 1, -1, 2, \) and \( -2 \).

Step 4: Write the Polynomial in Factored Form

Now that all zeros are identified, express the polynomial using its zeros. The factored form of \( P(x) \) is \( (x - 1)(x + 1)(x - 2)(x + 2) \), which represents all the rational zeros. By expanding this factored form, it confirms it corresponds to the original polynomial \( P(x) = x^{4} - 5x^{2} + 4 \).

Key Concepts

Synthetic Division

Synthetic division is a simplified form of polynomial division, especially useful for dividing a polynomial by a linear factor of the form \( x - c \). This method is particularly efficient when testing for possible rational zeros of a polynomial. For a polynomial \( P(x) \), you start by setting up a synthetic division tableau.

• Write down the coefficients of the polynomial in descending order of powers.
• Choose a potential zero \( c \) from the rational candidates identified through the Rational Root Theorem.
• Bring down the first coefficient as is.
• Multiply this number by \( c \), and add the result to the next coefficient. Repeat this process until you reach the last coefficient.
• If the remainder is zero, \( x = c \) is a root of the polynomial.

This process makes it easy to test multiple rational candidates without performing full polynomial division each time. It's efficient and reduces computational complexity.

Rational Zeros

The search for rational zeros of a polynomial can often be simplified using the Rational Root Theorem. This theorem is invaluable because it gives us a finite list of possible rational roots to test. According to this theorem:

• If a polynomial has any rational zeros, they are of the form \( \frac{p}{q} \), where:
• \( p \) is a factor of the constant term.
• \( q \) is a factor of the leading coefficient.
• For our polynomial \( P(x) = x^{4} - 5x^{2} + 4 \), the constant term is 4 and the leading coefficient is 1.
• This gives the possible rational zeros as \( \pm 1, \pm 2, \pm 4 \).

Once potential zeros are calculated, each can be verified using synthetic division to determine whether they are indeed zeros (roots) of the polynomial.

Polynomial Factoring

Factoring polynomials involves breaking them down into a product of simpler polynomials, which often reveals the roots or zeros of the polynomial. After identifying all the zeros of a polynomial using techniques like synthetic division, these zeros can be utilized to write the polynomial in its factored form.

• Consider the polynomial \( P(x) = x^{4} - 5x^{2} + 4 \).
• After determining the rational zeros, we know they are \( x = 1, -1, 2, -2 \).
• The polynomial can be expressed as \( (x - 1)(x + 1)(x - 2)(x + 2) \).

Factoring not only confirms the found roots but provides insight into the structure of the polynomial. The verification step of expanding this product back into its standard form confirms its equivalence with the original polynomial, demonstrating the validity of the entire process.