Problem 61 - Precalculus Mathematics for Calculus

Question

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(a)\) $$P(x)=x^{4}-5 x^{3}+6 x^{2}+4 x-8$$

Step-by-Step Solution

Answer

The real zeros are \( x = -1, x = 2, x = 2 \).

Step 1: Check for Rational Roots

To check for potential rational roots, we can apply the Rational Root Theorem, which suggests that any rational root is a factor of the constant term (in this case, -8) divided by a factor of the leading coefficient (which is 1). The potential rational roots are \( \pm 1, \pm 2, \pm 4, \pm 8 \).

Step 2: Test Rational Roots

We test each of the rational roots to see if it makes the polynomial zero.For example, by substituting \( x = 1 \) into \( P(x) \):\[P(1) = (1)^4 - 5(1)^3 + 6(1)^2 + 4(1) - 8 = 1 - 5 + 6 + 4 - 8 = -2\]We repeat similar calculations for \( x = -1, 2, -2, 4, -4, 8, -8 \), finding that \( x = 2 \) is a root, because:\[P(2) = (2)^4 - 5(2)^3 + 6(2)^2 + 4(2) - 8 = 16 - 40 + 24 + 8 - 8 = 0\]

Step 3: Synthetic Division with Found Root

Since we have found that \( x=2 \) is a root, we perform synthetic division to divide the polynomial \( P(x) \) by \( x-2 \). This will simplify the polynomial and help us find other roots.Using synthetic division, we have:\[\begin{array}{r|rrrrr}2 & 1 & -5 & 6 & 4 & -8 \ & & 2 & -6 & 0 & 8 \\hline & 1 & -3 & 0 & 4 & 0 \\end{array}\]The quotient is \( x^3 - 3x^2 + 0x + 4 \).

Step 4: Solve the Quartic Polynomial (Factored Form)

We need to solve \( x^3 - 3x^2 + 4 = 0 \). We'll use factorization and possibly the quadratic formula.First, check simple factors:Since \( x=2 \) is already determined as a root, divide again by \( x-2 \) using synthetic division on \( x^3 - 3x^2 + 4 \):\[\begin{array}{r|rrrr}2 & 1 & -3 & 0 & 4 \ & & 2 & -2 & 4 \\hline & 1 & -1 & -2 & 0 \\end{array}\]Resulting in \( x^2 - x - 2 \).

Step 5: Factor the Quadratic Equation

Now solve the quadratic equation \( x^2 - x - 2 = 0 \). This can be factored as:\[(x-2)(x+1) = 0\]Setting each factor to zero gives us the roots \( x = 2 \) and \( x = -1 \).

Step 6: Write All Real Zeros

From our work, the real zeros of \( P(x) \) are \( x = 2 \) (with multiplicity 2) and \( x = -1 \). So, the zeros are \( x = 2, x = 2, x = -1 \).

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool for finding the roots of quadratic equations, which are polynomials of the form \( ax^2 + bx + c = 0 \). By using this formula, you can find the values of \( x \) that make the equation true. Here's the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]To use the quadratic formula:

• Identify the coefficients \( a \), \( b \), and \( c \) from your quadratic equation.
• Substitute these values into the formula.
• Solve for \( x \) using basic arithmetic operations.

The term \( b^2 - 4ac \) is called the discriminant. It determines the nature of the roots:

• If \( b^2 - 4ac > 0 \), you'll have two distinct real roots.
• If \( b^2 - 4ac = 0 \), there's exactly one real root (a repeated root).
• If \( b^2 - 4ac < 0 \), the roots are complex and not real numbers.

In our example, once synthetic division has simplified the polynomial to \( x^2 - x - 2 = 0 \), we could use the quadratic formula if necessary, but here it is possible to factor directly for simplicity.

Rational Root Theorem

The Rational Root Theorem is a nifty shortcut in polynomial math that helps predict which rational numbers might be roots of a polynomial. If a polynomial with integer coefficients has a rational root, \( \frac{p}{q} \), then:

• \( p \) is a factor of the constant term of the polynomial.
• \( q \) is a factor of the leading coefficient.

In our problem, the polynomial \( P(x) = x^4 - 5x^3 + 6x^2 + 4x - 8 \) has a constant term of \(-8\) and a leading coefficient of \(1\). Therefore, potential rational roots are the factors of \(-8\). These are \( \pm 1, \pm 2, \pm 4, \pm 8 \). These potential roots are then tested in the polynomial to see if they make it zero. This initially identified \( x = 2 \) as a root.

Synthetic Division

Synthetic division is a simplified, efficient method for dividing polynomials, especially handy when testing potential roots found using the Rational Root Theorem.Here's how it's typically done:

• Write down the coefficients of the polynomial you wish to divide.
• Use the identified root. For example, if you suspect \( x = c \) is a root, you'll use \( x - c \) as your divisor.
• Perform the synthetic division process, bringing down numbers according to synthetic division steps.

In our exercise:

• We use the root \( x = 2 \) to divide \( P(x) = x^4 - 5x^3 + 6x^2 + 4x - 8 \).
• This simplifies the polynomial to \( x^3 - 3x^2 + 0x + 4 \).
• Synthetic division is applied again with \( x = 2 \) to further reduce the polynomial to a simpler quadratic equation \( x^2 - x - 2 \).

This method provides an effective way to break down complex polynomials into simpler components, making it easier to find all potential roots.