Problem 29 - Algebra and Trigonometry

Question

Find all rational zeros of the polynomial. $$ P(x)=4 x^{3}+4 x^{2}-x-1 $$

Step-by-Step Solution

Answer

The rational zeros are \(-1, \pm \frac{1}{2}\).

Step 1: Understand the Problem

We need to find the rational zeros of the polynomial \(P(x)=4x^3+4x^2-x-1\). Rational zeros, if any, can be found using the Rational Root Theorem.

Step 2: Apply the Rational Root Theorem

According to the Rational Root Theorem, any rational zero \(\frac{p}{q}\) of the polynomial is such that \(p\) (a factor of the constant term -1) and \(q\) (a factor of the leading coefficient 4).

Step 3: List Possible Rational Zeros

The factors of -1 are \(\pm 1\), and the factors of 4 are \(\pm 1, \pm 2, \pm 4\). Therefore, the possible rational zeros are \(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}\).

Step 4: Evaluate the Polynomial

Substitute each possible rational zero into \(P(x)\) to check if \(P(x)=0\). Calculate \(P(1), P(-1), P(\frac{1}{2}), P(-\frac{1}{2}), P(\frac{1}{4}),\) and \(P(-\frac{1}{4})\).

Step 5: Check \(P(1)\)

\(P(1) = 4(1)^3+4(1)^2-(1)-1 = 4+4-1-1 = 6\). Not a zero.

Step 6: Check \(P(-1)\)

\(P(-1) = 4(-1)^3+4(-1)^2-(-1)-1 = -4+4+1-1 = 0\). This is a zero.

Step 7: Refine the Search

Since \(x+1\) is a factor of \(P(x)\), perform synthetic division of \(P(x)\) by \(x+1\) to find the other factors.

Step 8: Synthetic Division

Divide the polynomial by \(x+1\) using synthetic division: - Coefficients: 4, 4, -1, -1- Bring down the 4- Multiply by -1 and add to next coefficient: - Next term: 4 - 4 = 0 - Next term: 0 - 1 = -1 - Next term: -1 - 1 = -2 (remainder)This leaves us with \(4x^2 + 0x - 1\) after dividing.

Step 9: Solve Quadratic \(4x^2 - 1 = 0\)

The remaining quadratic equation is \(4x^2 - 1 = 0\). Solve for \(x\) by setting \(4x^2 - 1 = 0\), leading to \(4x^2 = 1\) and \(x^2 = \frac{1}{4}\). Thus, \(x = \pm \frac{1}{2}\).

Step 10: Conclude Rational Zeros

The rational zeros of the polynomial \(P(x) = 4x^3+4x^2-x-1\) are \(x = -1, \pm \frac{1}{2}\).

Key Concepts

Rational Zeros

When tackling polynomial equations, one key step is finding its rational zeros. Rational zeros are values of \(x\) that make the polynomial equal to zero, and they can be expressed as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers. Learning about rational zeros is crucial for simplifying and solving polynomial expressions efficiently.

To determine possible rational zeros, we leverage the Rational Root Theorem, a fundamental tool in algebra. The theorem states that any potential rational zero \(\frac{p}{q}\) must have its numerator \(p\) as a factor of the constant term (the last term of the polynomial), and its denominator \(q\) as a factor of the leading coefficient (the coefficient of the term with the highest degree).

For instance, in the polynomial \(P(x) = 4x^3 + 4x^2 - x - 1\), the constant term is \(-1\), and the leading coefficient is \(4\). Thus, the factors of \(-1\) and \(4\) give us the possible rational zeros, which need to be tested to verify if any satisfy \(P(x) = 0\). This strategic process greatly narrows down potential solutions, making the task of finding zeros more manageable.

Synthetic Division

Synthetic division is an efficient shortcut method used for dividing polynomials, particularly when dealing with polynomials of a higher degree. It simplifies the division process significantly compared to long division of polynomials. This method is most effective when dividing by linear expressions such as \(x - c\).

To use synthetic division, align the coefficients of the polynomial in descending order of the degree. The divisor \(c\) is derived from the factor \(x - c\). Let's see it in action with our polynomial example \(P(x) = 4x^3 + 4x^2 - x - 1\), when divided by \(x + 1\) (where \(c = -1\)).

The process involves:

• Writing down the coefficients: \(4, 4, -1, -1\).
• Bringing the leading coefficient down as is.
• Multplying this coefficient by \(c\) and adding it to the next coefficient, continuing this step till the end.

After performing synthetic division, what's left is a polynomial with one degree less, which can be further analyzed or factored. In our exercise, synthetic division confirmed that \(x + 1\) is indeed a factor, helping us identify a simpler quadratic equation, \(4x^2 - 1\), to solve.

Polynomial Factorization

Factorization of polynomials is about breaking down a complex polynomial into simpler polynomials that multiply together to give the original polynomial. This step is crucial in finding polynomial solutions, as once factored, it's much easier to identify the zeros.

Let's revisit our problem polynomial \(P(x) = 4x^3 + 4x^2 - x - 1\) after identifying and isolating one factor, \(x + 1\), through synthetic division. The remainder polynomial was \(4x^2 - 1\), indicating it's time to factor again.

For quadratics like \(4x^2 - 1\), factorization into simpler polynomials can often be done by recognizing patterns like the difference of squares:

• \(4x^2 - 1 = (2x)^2 - 1^2\)
• Therefore, it can be factored into \((2x - 1)(2x + 1)\).

Now, our original cubic polynomial is fully factored as \((x + 1)(2x - 1)(2x + 1)\). Each factor represents a potential zero, thus solving these will yield the rational zeros \(x = -1, \frac{1}{2}, -\frac{1}{2}\). Recognizing factored forms readily increases efficiency in solving polynomial problems.