Problem 59 - Precalculus Mathematics for Calculus

Question

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(a)\) $$P(x)=2 x^{3}-7 x^{2}+4 x+4$$

Step-by-Step Solution

Answer

The real zeros are \(x = 1\), \(x = \frac{5 + \sqrt{57}}{4}\), and \(x = \frac{5 - \sqrt{57}}{4}\).

Step 1: Identify the Polynomial Coefficients

The polynomial is given as \(P(x) = 2x^3 - 7x^2 + 4x + 4\). From this polynomial, identify the coefficients: \(a = 2\), \(b = -7\), \(c = 4\), and \(d = 4\). These coefficients will be used for finding the zeros.

Step 2: Perform Synthetic Division to Find a Possible Root

Use the Rational Root Theorem to list possible rational roots, which are \(\pm 1, \pm 2, \pm 4, \pm \frac{1}{2}\). Test them using synthetic division to find a root. Upon testing, \(x = 1\) is found to be a root since the remainder of the synthetic division is zero.

Step 3: Factor the Polynomial Using the Found Root

Since \(x = 1\) is a root, factor \(x - 1\) out of \(P(x)\). Divide \(2x^3 - 7x^2 + 4x + 4\) by \(x - 1\) using synthetic division, resulting in the quotient \(2x^2 - 5x - 4\). So, \(P(x) = (x - 1)(2x^2 - 5x - 4)\).

Step 4: Solve the Quadratic Equation

Solve the quadratic equation \(2x^2 - 5x - 4 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 2\), \(b = -5\), \(c = -4\). Calculate the discriminant: \(b^2 - 4ac = (-5)^2 - 4(2)(-4) = 25 + 32 = 57\). Thus, the roots are \[x = \frac{5 \pm \sqrt{57}}{4}\].

Step 5: List All Real Zeros

Combine the previously found root \(x = 1\) and the roots from the quadratic: \(x = \frac{5 + \sqrt{57}}{4}\) and \(x = \frac{5 - \sqrt{57}}{4}\). Therefore, the real zeros of the polynomial are \(x = 1\), \(x = \frac{5 + \sqrt{57}}{4}\), and \(x = \frac{5 - \sqrt{57}}{4}\).

Key Concepts

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a linear binomial of the form \(x - c\). This method is especially useful for evaluating possible roots of a polynomial. Begin by writing down the coefficients of the polynomial, and then use the potential root (from the Rational Root Theorem) in the synthetic division process.

Here, with the polynomial \(P(x) = 2x^3 - 7x^2 + 4x + 4\) and potential root \(x = 1\), you will:
\[\begin{array}{r|rrr} 1 & 2 & -7 & 4 & 4 \ & & 2 & -5 & -1 \ \hline & 2 & -5 & -1 & 0 \end{array}\]

The numbers below the line are the coefficients of the quotient polynomial. Because the last number is 0, it confirms that \(x = 1\) is indeed a root. As a result, synthetic division greatly simplifies the process of factoring and finding zeros of the polynomial.

Rational Root Theorem

The Rational Root Theorem is a handy tool for finding possible rational zeros of a polynomial with integer coefficients. It states that any potential rational root, expressed as \(\frac{p}{q}\), must have \(p\) as a factor of the constant term and \(q\) as a factor of the leading coefficient.

For the polynomial \(P(x) = 2x^3 - 7x^2 + 4x + 4\), the constant term is 4 and the leading coefficient is 2. Thus, the possible rational roots are factors of 4 divided by factors of 2:

• Factors of 4: \(\pm 1, \pm 2, \pm 4\)
• Factors of 2: \(\pm 1, \pm 2\)
• Possible rational roots: \(\pm 1, \pm 2, \pm 4, \pm \frac{1}{2}\)

Utilize synthetic division to test these possible roots, as was done in the exercise where \(x = 1\) was confirmed as a root.

Quadratic Formula

The quadratic formula is vital for finding zeros of a quadratic equation of the form \(ax^2 + bx + c = 0\). It is given by:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

This formula allows for finding both real and complex roots, depending on the value of the discriminant \(b^2 - 4ac\). If the discriminant is positive, as in the polynomial \(2x^2 - 5x - 4\), two real roots exist:

• Discriminant: \((-5)^2 - 4 \cdot 2 \cdot (-4) = 25 + 32 = 57\)
• Roots: \[x = \frac{5 \pm \sqrt{57}}{4}\]

The quadratic formula is very effective when a quadratic factor results from polynomial factorization.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into simpler polynomials that, when multiplied together, give the original polynomial. Once a root is found using synthetic division or the Rational Root Theorem, the polynomial can be factored further.

In the exercise, after confirming \(x = 1\) as a root, the process of dividing the polynomial by \(x - 1\) yields a quadratic \(2x^2 - 5x - 4\). This step represents the factorization of the cubic polynomial into:

\[P(x) = (x - 1)(2x^2 - 5x - 4)\]

Solving this quadratic factor using the quadratic formula gives you the remaining zeros of the polynomial. Factorization reduces complex polynomials to simpler terms and highlights both real and complex roots effectively.