Problem 63 - A Graphical Approach to College Algebra

Question

For each polynomial, at least one zero is given. Find all others analytically. $$P(x)=x^{3}-2 x+1 ; 1$$

Step-by-Step Solution

Answer

The zeros are \(x = 1\), \(x = \frac{-1 + \sqrt{5}}{2}\), and \(x = \frac{-1 - \sqrt{5}}{2}\).

Step 1: Verify the Given Zero

We need to verify that 1 is indeed a zero of the polynomial \(P(x) = x^3 - 2x + 1\). We do this by substituting \(x = 1\) into the polynomial:\[P(1) = 1^3 - 2(1) + 1 = 1 - 2 + 1 = 0.\]Since \(P(1) = 0\), \(x = 1\) is a valid zero.

Step 2: Perform Polynomial Division

Having confirmed that \(x = 1\) is a zero, use polynomial division to divide \(P(x)\) by \(x - 1\). Performing the synthetic division gives:1 | 1 0 -2 1 | 1 1 -1 ---------------- 1 1 -1 0This division shows that \(P(x) = (x - 1)(x^2 + x - 1)\).

Step 3: Find Zeros of the Quadratic Factor

Now we need to find the zeros of the quadratic \(x^2 + x - 1\). To do this, use the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\]where \(a = 1\), \(b = 1\), and \(c = -1\).Calculate the discriminant:\[b^2 - 4ac = 1^2 - 4(1)(-1) = 1 + 4 = 5.\]Find the roots:\[x = \frac{-1 \pm \sqrt{5}}{2}.\]

Step 4: State All Zeros of the Polynomial

The zeros of the polynomial \(P(x)\) are \(x = 1\) and \[x = \frac{-1 + \sqrt{5}}{2}, \frac{-1 - \sqrt{5}}{2}.\]

Key Concepts

Polynomial Division

Polynomial division is a method used to divide polynomials, similar to how regular long division works, but with variables. In the context of this exercise, we are using it to find other factors of a polynomial after identifying one of its zeros.

Here's how it works:

• Identify the zero of the polynomial. In this problem, it's given as 1.
• Use the zero to create a binomial, which in this case is \(x - 1\).
• Divide the original polynomial \(P(x) = x^3 - 2x + 1\) by \(x - 1\). This is done using synthetic or long division.

After dividing, you uncover the other factor of the polynomial. In this example, dividing by \(x - 1\) results in the quotient \(x^2 + x - 1\). This quotient is a simpler polynomial whose zeros we can find next.

Quadratic Formula

The quadratic formula is a useful tool that helps us find the roots of a quadratic equation, which is any polynomial of the form \(ax^2 + bx + c = 0\).

For the exercise at hand, after dividing the original polynomial, you're left with a quadratic polynomial \(x^2 + x - 1\). To find its zeros, you set it equal to zero and apply the quadratic formula:

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

Here, \(a = 1\), \(b = 1\), and \(c = -1\). Substituting these values into the formula will help you find the values of \(x\) that satisfy this equation, revealing the zeros of the quadratic component of the polynomial.

Discriminant Calculation

The discriminant is a part of the quadratic formula under the square root, denoted by \(b^2 - 4ac\). It's crucial because it tells you the nature of the roots of the quadratic equation.

For the quadratic \(x^2 + x - 1\), compute the discriminant as follows:

• Calculate \(b^2\): \(1^2 = 1\)
• Calculate \(4ac\): \(4 \times 1 \times (-1) = -4\)
• Add the results: \(1 - (-4) = 5\)

A positive discriminant, as here being 5, indicates there are two distinct real roots. The existence of distinct real roots is confirmed because the quadratic formula yields two different results when this positive discriminant is plugged in.