Problem 32 - Precalculus Mathematics for Calculus

Question

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=2 x^{3}+7 x^{2}+4 x-4$$

Step-by-Step Solution

Answer

The rational zeros are \( x = -1, \frac{1}{2}, -4 \). The factored form is \((x + 1)(2x - 1)(x + 4)\)."

Step 1: Identifying Possible Rational Zeros

First, use the Rational Root Theorem to determine all possible rational zeros of the polynomial. According to this theorem, any rational zero, expressed as \( \frac{p}{q} \), has a numerator \( p \) that is a factor of the constant term (-4) and a denominator \( q \) that is a factor of the leading coefficient (2). Thus, possible rational zeros are \( \pm 1, \pm 2, \pm 4, \pm \frac{1}{2}, \pm \frac{1}{4} \).

Step 2: Testing for Rational Zeros

Now, test each possible rational zero using synthetic division or direct substitution into the polynomial \( P(x) \). Start with \( x = 1 \). Substitute into the polynomial: \( P(1) = 2(1)^3 + 7(1)^2 + 4(1) - 4 = 9 \). Since it is not zero, \( x = 1 \) is not a root. Repeat this process for other possible rational roots.

Step 3: Finding a Valid Zero

Upon testing, you'll find that substituting \( x = -1 \) in the polynomial gives \( P(-1) = 2(-1)^3 + 7(-1)^2 + 4(-1) - 4 = 0 \). Thus, \( x = -1 \) is a rational root.

Step 4: Performing Synthetic Division

Use synthetic division to divide the polynomial by \( x + 1 \) (since \( x = -1 \) is a root), which simplifies the polynomial. Perform synthetic division with the coefficients: 2, 7, 4, and -4. After division, the quotient yields \( 2x^2 + 5x - 4 \).

Step 5: Factoring the Quotient

Next, factor the resulting quadratic polynomial \( 2x^2 + 5x - 4 \). This can be factored by grouping or by trial and error to find that it factors to \((2x - 1)(x + 4)\).

Step 6: Writing the Polynomial in Factored Form

Combine the factor found in synthetic division with the factored quadratic form. Thus, the original polynomial \( P(x) = 2x^3 + 7x^2 + 4x - 4 \) can be written as \((x + 1)(2x - 1)(x + 4)\).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a handy tool when dealing with polynomials, especially when trying to find possible rational zeros effortlessly. It tells us that any rational zero of a polynomial, written as \( \frac{p}{q} \), has a numerator \( p \) that is a factor of the constant term and a denominator \( q \) that is a factor of the leading coefficient. This means you only need to test a handful of values instead of guessing.

For example, in the polynomial \( P(x) = 2x^3 + 7x^2 + 4x - 4 \), the constant term is \(-4\) and the leading coefficient is \(2\). Therefore, possible choices for \( p \) are \( \pm 1, \pm 2, \pm 4 \), and for \( q \), they are \( \pm 1, \pm 2 \).
Thus, our possible rational zeros could be \( \pm 1, \pm 2, \pm 4, \pm \frac{1}{2}, \pm \frac{1}{4} \). You can dramatically narrow down the search for roots using this approach.

In summary, by applying the Rational Root Theorem, finding potential rational zeros becomes much less daunting, guiding us to a more systematic way of solving polynomials.

synthetic division

Synthetic division is a simplified form of polynomial division, one that saves time and reduces the complexity involved in long division. It works by transforming the process into easy-to-manage calculations on the coefficients of the polynomial.

After finding a rational zero using the Rational Root Theorem, synthetic division helps in simplifying the polynomial further. Let's consider our polynomial, \( P(x) = 2x^3 + 7x^2 + 4x - 4 \), and assume through testing that \( x = -1 \) is a zero. We now perform synthetic division using this root, dividing the polynomial by \( x + 1 \).

• Align the coefficients: 2, 7, 4, -4 horizontally.
• Bring down the first coefficient, 2.
• Multiply by the root, -1, and add to the next coefficient.
• Continue until you've used all terms.

Upon following these steps, you find a new, simpler polynomial, \( 2x^2 + 5x - 4 \). Synthetic division not only helps simplify calculations but also confirms you've found a legitimate root.

factoring polynomials

Factoring polynomials is the final puzzle piece in breaking down a polynomial into its simplest form, making it easier to solve. Once you have found one of the roots through synthetic division, and reduced the polynomial's degree, your task is to factor the remaining polynomial.

In our example, after using synthetic division on \( P(x) = 2x^3 + 7x^2 + 4x - 4 \) with the root \( x = -1 \), you end up with the quadratic \( 2x^2 + 5x - 4 \).

Factoring this requires finding two binomials, which multiply back to give the quadratic. You can do this through test and trial, or by sketching a multiplication process to ensure your binomials are correct. For \( 2x^2 + 5x - 4 \), factorization results in \((2x - 1)(x + 4)\).

Finally, combine this with the binomial from the prior division step, \( x + 1 \), giving the complete factorization of the polynomial: \((x + 1)(2x - 1)(x + 4)\). This simplifies solving polynomial equations, making it easier to identify solutions or further analyze the behavior of the polynomial over different values.