Problem 41 - Precalculus Mathematics for Calculus

Question

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=2 x^{4}-7 x^{3}+3 x^{2}+8 x-4$$

Step-by-Step Solution

Answer

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

Step 1: Identify possible rational zeros

To find possible rational zeros, use the Rational Root Theorem, which states that any rational zero, \( \frac{p}{q} \), is such that \( p \) is a factor of the constant term \(-4\), and \( q \) is a factor of the leading coefficient \(2\). Thus, possible rational zeros are \( \pm 1, \pm 2, \pm 4, \pm \frac{1}{2}, \pm \frac{2}{2} (\text{i.e., } \pm1), \pm \frac{4}{2} (\text{i.e., } \pm2) \). Thus we have \( \pm 1, \pm 2, \pm 4, \pm \frac{1}{2} \) as possible zeros.

Step 2: Test possible rational zeros using synthetic division

Test the possible zeros using synthetic division. Start with \(x = 1\). Use synthetic division among all potential rational zeros until we find a zero. Testing \(x = 1\): \(2, -7, 3, 8, -4\) gives a non-zero remainder. Testing \(x = 2\) results in no remainder, so \(x = 2\) is a zero. We divide the polynomial by \(x - 2\), resulting in \(2x^3 - 3x^2 - 3x + 2\).

Step 3: Repeat zero finding for the quotient polynomial

Now, repeat the process for the polynomial \(2x^3 - 3x^2 - 3x + 2\) obtained from division. Try \(x = 1\) with synthetic division: the remainder is zero, so \(x=1\) is a root; we thus have \((x-1)\) being another factor, and the polynomial becomes \(2x^2 - x - 2\).

Step 4: Factor the remaining quadratic expression

Factor the quadratic \(2x^2 - x - 2\). We find two numbers that multiply to \(2(-2) = -4\) and add to \(-1\), which are \(-2\) and \(1\). Thus, we can factor this quadratic as \((2x^2 + 2x) - (x + 2)\). Factor by grouping to get \((2x + 1)(x - 2)\).

Step 5: Write the polynomial in factored form

The full factorization of \(P(x)\) becomes \((x - 2)^2(x - 1)(2x + 1)\). Here, we include \((x - 2)\) twice because it corresponds to two successful synthetic division zeros.

Key Concepts

Polynomial Factoring

Polynomial factoring is the process of breaking down a polynomial into simpler polynomials called factors. These factors, when multiplied together, give you the original polynomial. Factoring is a key skill needed for solving polynomial equations, as it helps with finding the roots or zeros of the polynomial.

To factor a polynomial effectively:

• First, look for a greatest common factor (GCF) that might simplify the polynomial by removing common terms.
• Next, use various factoring techniques such as grouping, special product rules, or trial and error to break it down.
• Once the polynomial is in its simplest form, its factors can reveal the polynomial's zeros, which are critical for solving equations.

Factoring can appear daunting, but breaking down the process into manageable steps makes it easier to handle. For instance, our example polynomial eventually factors fully as \((x - 2)^2(x - 1)(2x + 1)\). Each factor corresponds to a root of the polynomial.

Synthetic Division

Synthetic division is a simplified form of polynomial division, specifically used when dividing by a linear factor of the form \(x - c\). It is especially useful for quickly checking whether potential rational zeros, suggested by the Rational Root Theorem, are actual zeros of a polynomial.

In the exercise, synthetic division was used to test potential zeros like \(x = 2\), verifying it yielded no remainder, confirming it's a root. This division not only confirms roots but also simplifies the polynomial tack for solving or further factorization.

To use synthetic division:

• Take the coefficients of your polynomial and arrange them in order.
• Bring down the first coefficient to start your row of results.
• Multiply the divisor root by the number just brought down and add it to the next coefficient.
• Continue this process across all coefficients.
• If you end with a zero remainder, you've found a factor \(x - c\).

Synthetic division, though seeming intricate at first, drastically cuts down the work of finding polynomial zeros.

Quadratic Factorization

Quadratic factorization refers to the process of breaking a quadratic equation into two simpler binomial expressions. Solving quadratics by factoring is one of several methods and revolves around finding two numbers that multiply to make the quadratic's constant term while adding to make its linear coefficient.

For example, in our problem, the quadratic \(2x^2 - x - 2\) was factored into \((2x + 1)(x - 2)\). This factorization step involved seeking two numbers, \(-2\) and \(1\), which multiply to \(-4\) and add to \(-1\).

When factoring quadratics:

• Attempt to split the middle term accurately into integers that fulfill both multiplication and addition criteria.
• Use methods like factoring by grouping if the terms resist straightforward decomposition.
• Check the factored form by expanding to ensure it returns to the original polynomial.

Quadratic factorization is crucial for finding rational zeros, as its factors directly yield roots of the polynomial equation.

Rational Zeros

Rational zeros are solutions of a polynomial equation represented as fractions or whole numbers. The Rational Root Theorem is the primary tool for determining these potential zeros. It states that any rational zero \(\frac{p}{q}\) of a polynomial is such that \(p\) is a factor of the constant term, while \(q\) is a factor of the leading coefficient.

In the given polynomial \(P(x)=2x^{4}-7x^{3}+3x^{2}+8x-4\), the possible rational zeros were extracted as \(\pm 1, \pm 2, \pm 4, \pm \frac{1}{2}\) using this theorem.

To identify rational zeros:

• List all factors of the constant term.
• List all factors of the leading coefficient.
• Form all combinations of these factors as potential zeros.
• Use synthetic division to verify which combinations are actual zeros.

Understanding rational zeros paves the way for efficiently solving polynomial equations and brings clarity to polynomial structures.