Problem 22 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The rational roots are 1, 2, and -2.

Step 1: State the Rational Root Theorem

The Rational Root Theorem suggests that any rational root of the polynomial \( P(x) = x^4 - 2x^3 - 3x^2 + 8x - 4 \) will be a fraction \( \frac{p}{q} \) where \( p \) is a factor of the constant term, -4, and \( q \) is a factor of the leading coefficient, 1.

Step 2: Identify the Possible Rational Roots

For this polynomial, the possible values of \( p \) (factors of -4) are \( \pm 1, \pm 2, \pm 4 \). Since the leading coefficient is 1, \( q = \pm 1 \). Thus, the possible rational roots are \( \pm 1, \pm 2, \pm 4 \).

Step 3: Test Each Possible Root Using Synthetic Division

Perform synthetic division for each possible rational root. Substitute each value into the polynomial and check if it results in a remainder of 0, indicating it is a root.

Step 4: Verify -1 as a Root

Perform synthetic division with \( x = -1 \):\[\begin{array}{r|rrrrr}-1 & 1 & -2 & -3 & 8 & -4 \ & & -1 & 3 & 0 & -8 \\hline & 1 & -3 & 0 & 8 & -12 \\end{array}\]The remainder is not 0, so -1 is not a root.

Step 5: Verify 1 as a Root

Perform synthetic division with \( x = 1 \):\[\begin{array}{r|rrrrr}1 & 1 & -2 & -3 & 8 & -4 \ & & 1 & -1 & -4 & 4 \\hline & 1 & -1 & -4 & 4 & 0 \\end{array}\]The remainder is 0, so 1 is a root.

Step 6: Factor the Polynomial with x-1

Since \( x = 1 \) is a root, divide the polynomial \( P(x) \) by \( x - 1 \) using the synthetic result:\( x^3 - x^2 - 4x + 4 \).

Step 7: Find Roots of the Quotient Polynomial

We now need to find the rational roots of \( x^3 - x^2 - 4x + 4 \). We repeat the rational root theorem process on this polynomial with possible roots: \( \pm 1, \pm 2, \pm 4 \).

Step 8: Verify 2 as a Root for \( x^3 - x^2 - 4x + 4 \)

Perform synthetic division with \( x = 2 \):\[\begin{array}{r|rrrr}2 & 1 & -1 & -4 & 4 \ & & 2 & 2 & -4 \\hline & 1 & 1 & -2 & 0 \\end{array}\]The remainder is 0, so 2 is a root.

Step 9: Factor the Quotient Polynomial Further

Since 2 is a root, factor \( x^3 - x^2 - 4x + 4 \) by \( x - 2 \) to get \( x^2 + x - 2 \).

Step 10: Solve the Quadratic Polynomial

The quadratic \( x^2 + x - 2 \) can be solved by factoring:\((x - 1)(x + 2) = 0\).Thus, the additional roots are \( x = 1 \) and \( x = -2 \).

Step 11: Compile All Rational Roots

The rational roots of the original polynomial \( P(x) = x^4 - 2x^3 - 3x^2 + 8x - 4 \) are \( 1, 2, \) and \( -2 \).

Key Concepts

Polynomial Division

Polynomial division is a method similar to long division but is used specifically for dividing polynomials.
It helps to simplify polynomial expressions and find the roots of polynomials as well. To perform polynomial division, follow these steps:

• Arrange the terms in descending order of the degrees.
• Identify the highest degree term in the dividend and the divisor.
• Divide the first term of the dividend by the first term of the divisor.
• Multiply the entire divisor by this result and subtract from the current dividend.
• Repeat the process with the remainder until it becomes lower in degree than the divisor.

This method gives the quotient and the remainder, helping to simplify a polynomial for further analysis of its roots.

Synthetic Division

Synthetic division is a simplified version of polynomial division primarily used for evaluating polynomials at certain points, or finding their roots. It involves fewer steps and calculations, making it quicker and simpler than long division of polynomials. In synthetic division:

• Only the coefficients of the polynomials are used.
• The divisor should be of the form \( x - c \), where \( c \) is a constant.
• The process involves 'bringing down', multiplying, and adding values within a structured grid setup.
• The final result gives a quotient polynomial and a remainder.

This method is very useful for determining whether a given number is a root of a polynomial by checking if the remainder is zero.

Quadratic Factorization

Quadratic factorization involves breaking down a quadratic polynomial into simpler binomial factors. This process is essential in solving quadratic equations since it directly shows the roots of the polynomial. To factorize a quadratic polynomial like \( ax^2 + bx + c \):

• Look for two numbers whose product is \( ac \) and sum is \( b \).
• Use these numbers to break the middle term into two separate terms.
• Group the terms in pairs and factor out the greatest common factor from each pair.
• Combine these to achieve a product of binomials.

For example, \( x^2 + x - 2 \) factored becomes \((x - 1)(x + 2)\), revealing its roots \( x = 1 \) and \( x = -2 \).

Possible Rational Roots

The Rational Root Theorem is a key tool to identify possible rational roots of a polynomial equation.According to this theorem, any potential rational root of a polynomial with integer coefficients is of the form \( \frac{p}{q} \), where:

• \( p \) is a factor of the constant term of the polynomial.
• \( q \) is a factor of the leading coefficient of the polynomial.

Once the possible rational roots \( \pm 1, \pm 2, \pm 4 \) are identified for the polynomial \( x^4 - 2x^3 - 3x^2 + 8x - 4 \), these are tested using synthetic division to discover any actual roots of the polynomial, which were found to be \( 1, 2, \) and \( -2 \).