Problem 33 - Algebra and Trigonometry

Question

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$ P(x)=4 x^{3}+4 x^{2}-x-1 $$

Step-by-Step Solution

Answer

The rational zeros are \(-1, \frac{1}{2}, \text{and} -\frac{1}{2}\); the polynomial factors as \((x + 1)(2x - 1)(2x + 1)\).

Step 1: Factor Theorem Introduction

We can find rational zeros of a polynomial using the Factor Theorem, which states that if \( c \) is a root of polynomial \( P(x) \), then \( (x - c) \) is a factor of \( P(x) \). This involves checking possible rational roots.

Step 2: List Possible Rational Zeros

According to the Rational Root Theorem, potential rational zeros are \( \pm \frac{p}{q} \), where \( p \) is a factor of the constant term (-1), and \( q \) is a factor of the leading coefficient (4). Therefore, the possible rational zeros are \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{4} \).

Step 3: Evaluate Possible Zeros

Test each possible zero using substitution into the polynomial \( P(x) \). Start with \( x = 1 \): \( P(1) = 4(1)^3 + 4(1)^2 - 1 - 1 = 6 \). Since \( P(1) eq 0 \), \( x = 1 \) is not a zero.

Step 4: Find Actual Rational Zero

Continue testing other candidates:- For \( x = -1\), \( P(-1) = 4(-1)^3 + 4(-1)^2 - (-1) - 1 = 0 \).- Therefore, \( x = -1 \) is a zero.

Step 5: Factor Out \((x + 1)\)

Since \( x = -1 \) is a zero, \((x + 1)\) is a factor of \( P(x)\). Use synthetic division to divide \( P(x) \) by \( x + 1 \). The result is \( 4x^2 - 1 \).

Step 6: Factor the Quadratic

The quotient \( 4x^2 - 1 \) is a difference of squares and can be factored as \((2x - 1)(2x + 1)\). This gives the complete factorization of the polynomial.

Step 7: Write the Polynomial in Factored Form

Combine the factors found: \( P(x) = (x + 1)(2x - 1)(2x + 1) \). All roots of the equation (zeros) are \( x = -1, x = \frac{1}{2}, \text{and} \ x = -\frac{1}{2} \).

Key Concepts

Polynomial Factorization

Polynomial factorization is the process of breaking a polynomial into simpler polynomials whose product is equal to the original polynomial. For example, consider the polynomial \( P(x) = 4x^3 + 4x^2 - x - 1 \). Our goal is to express this polynomial as the product of lower-degree polynomials.We start by finding a root using the Rational Root Theorem, which helps identify possible rational zeros that make the polynomial equal to zero.Once a zero is found, you can use synthetic division to simplify the polynomial further.Factorization not only simplifies the polynomial but also makes finding solutions to the polynomial equation easier. This simplifies calculations and provides insight into the polynomial's properties.By factoring \( P(x) \), we determine it as \((x + 1)(2x - 1)(2x + 1)\). So, factorization is a step-by-step puzzle that breaks down complex expressions.

Synthetic Division

Synthetic division is a quick and efficient method to divide a polynomial by a binomial of the form \(x - c\). It's a simpler alternative to the traditional polynomial long division. This technique is particularly useful for testing potential rational zeros. Let's relate it to our polynomial \( P(x) = 4x^3 + 4x^2 - x - 1 \). When we discovered that \( x = -1 \) was a zero, we used synthetic division to divide \( P(x) \) by \( x + 1 \), deriving the quotient \( 4x^2 - 1 \).Here's a brief overview of how synthetic division is performed:

• Write the coefficients of the polynomial in a row.
• Use the zero \( c \) outside the division symbol.
• Bring down the leading coefficient next to the constant below the line.
• Multiply \( c \) by this number, writing the product below the next coefficient.
• Add the column, and repeat the process until completion.

The process concludes by providing a simplified polynomial and a possible remainder. This remainder helps confirm whether the divisor term is indeed a factor.

Factor Theorem

The Factor Theorem is a fundamental concept in algebra that links roots of polynomials with their factors. It states that if \( c \) is a root of a polynomial \( P(x) \), then \((x - c)\) is a factor of \( P(x) \).To utilize the Factor Theorem, you generally follow these steps:

• Identify potential zeros using the Rational Root Theorem.
• Verify actual zeros by substituting back into the polynomial.
• Use these zeros to factor the polynomial.

In our example, we found that \( x = -1 \) was a root of \( P(x) \), hence \( (x + 1) \) is a factor. The Factor Theorem tells us that if a polynomial evaluates to zero for a particular value, then dividing the polynomial by the corresponding binomial will result in no remainder. This facilitates easily breaking down the polynomial into simpler components, as we did with \( 4x^2 - 1 \) after confirming \( x = -1 \) as a zero of the polynomial.

Difference of Squares

The difference of squares is a special factoring technique used to simplify expressions of the form \( a^2 - b^2 \). This technique utilizes the identity \( a^2 - b^2 = (a - b)(a + b) \). It's particularly useful when dealing with quadratic polynomials or when simplifying factors from higher-order polynomials.In the given problem, after using synthetic division, we identified the quadratic \( 4x^2 - 1 \).Notice that this expression can be seen as a difference of squares:- Here, \( 4x^2 \) is \((2x)^2\), and \( 1 \) is \( 1^2 \).Applying the difference of squares formula, we have:\[ 4x^2 - 1 = (2x)^2 - 1^2 = (2x - 1)(2x + 1) \]This allows us to factor the quadratic into simpler binomial terms efficiently. The difference of squares is an excellent tool for quickly factoring expressions and simplifying polynomial solutions.