Problem 23 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The rational zeros are \(1, -1, -3\), and the factored form is \((x - 1)(x + 1)(x + 3)\).

Step 1: Identify Possible Rational Zeros

To find the rational zeros of a polynomial, we use the Rational Root Theorem. For the polynomial \( P(x) = x^3 + 3x^2 - x - 3 \), possible rational zeros are the factors of the constant term (\(-3\)) divided by the factors of the leading coefficient (\(1\)). The possible rational zeros are: \( \pm 1, \pm 3 \).

Step 2: Test Possible Zeros Using Synthetic Division

Test each possible rational zero to see if it is an actual zero of the polynomial. Start with \(x=1\):1. Write the coefficients of the polynomial: \([1, 3, -1, -3]\).2. Use synthetic division for \(x - 1\): If the remainder is 0, then \(x = 1\) is a zero.3. Calculate: - Bring down the 1. - Multiply 1 by 1, add to 3 yields 4. - Multiply 1 by 4, add to -1 yields 3. - Multiply 1 by 3, add to -3 yields 0.- The remainder is 0, so \(x = 1\) is a zero.

Step 3: Factor the Polynomial using the Found Zero

Since \(x = 1\) is a zero, \(x - 1\) is a factor of \(P(x)\). Use the quotient from synthetic division, \(x^2 + 4x + 3\), to factor further.\[ P(x) = (x - 1)(x^2 + 4x + 3) \]

Step 4: Factor the Quadratic Polynomial

Factor \(x^2 + 4x + 3\) into \( (x + 1)(x + 3) \).This is done by finding two numbers that multiply to 3 and add to 4, which are 1 and 3.

Step 5: Verify the Complete Factored Form

Combine all factors to write \( P(x) \) in its fully factored form:\[ P(x) = (x - 1)(x + 1)(x + 3) \]Check by expanding to ensure it matches the original polynomial.

Key Concepts

Synthetic Division

Synthetic division is a quick and efficient method to divide polynomials when dividing by a linear factor, usually in the form of \(x - c\). This method is particularly useful when applying the Rational Root Theorem to test possible rational zeros. It simplifies the division process into a series of straightforward steps:

• Write down the coefficients of the polynomial. For our example, \(P(x) = x^3 + 3x^2 - x - 3\), the coefficients are \([1, 3, -1, -3]\).
• Use the suspected zero (like 1 in our case) to set up the synthetic division.
• Bring down the leading coefficient directly below the line.
• Multiply the number below the line by the suspect zero, and place the result in the next column. Add this result to the next coefficient above.

Continue this process until you finish all the coefficients. If the last number you compute (the remainder) is zero, then the suspect zero is an actual zero of the polynomial. This systematic approach significantly shortens potentially tedious calculations.

Factoring Polynomials

Factoring polynomials involves breaking down complex expressions into simpler, multiplied terms known as factors, which can be handled more easily. Once potential rational zeros have been identified, factoring becomes a way to express the polynomial using these zeros.
To factor polynomials:

• Find one or more zeros of the polynomial using techniques like the Rational Root Theorem and verify them using synthetic division or substitution.
• Use the zeros to create linear factors. For example, if \(x = 1\) is a zero, then \(x - 1\) is a linear factor.
• Repeat this process with a quotient polynomial, such as \(x^2 + 4x + 3\), continuing to factor further until no simpler terms are left.
• Combine all the factored components to present the polynomial in its fully factored form.

Mastering this method reveals deeper polynomial structures, makes graphing easier, and sets a foundation for solving polynomial equations.

Rational Root Theorem

The Rational Root Theorem is a powerful tool for finding the possible rational zeros (roots) of a polynomial. It states that any rational root, when in lowest terms \(\frac{p}{q}\), is such that \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient.
Applying the Rational Root Theorem includes these steps:

• Identify the constant term and the leading coefficient of the polynomial. For \(P(x) = x^3 + 3x^2 - x - 3\), these are \(-3\) and \(1\), respectively.
• List all possible factors of the constant term, which are \(\pm 1\) and \(\pm 3\).
• List all possible factors of the leading coefficient, which is just \(\pm 1\).
• Form possible rational roots by taking ratios of the constant term factors over the leading coefficient factors. Hence, the potential rational roots for our example are \(\pm 1\) and \(\pm 3\).

By pinpointing plausible candidates for real solutions, the theorem sets the stage for synthetic division and other tools to verify and further explore polynomial roots.