Problem 66 - A Graphical Approach to Precalculus with Limits

Question

For each polynomial at least one zero is given. Find all others analytically. $$P(x)=x^{3}-7 x^{2}+13 x-3 ; 3$$

Step-by-Step Solution

Answer

The zeros are \(x = 3\), \(x = 2 + \sqrt{3}\), and \(x = 2 - \sqrt{3}\).

Step 1: Use Given Zero

We start with the polynomial \(P(x) = x^3 - 7x^2 + 13x - 3\) and we know that \(x = 3\) is a zero of this polynomial. That means \((x - 3)\) is a factor of \(P(x)\).

Step 2: Perform Polynomial Division

We will perform synthetic division of \(P(x)\) by \((x - 3)\). Set up synthetic division by writing 3 (since 3 is the zero) and the coefficients of \(P(x)\): \(1, -7, 13, -3\).

Step 3: Execute Synthetic Division Algorithm

1. Bring down the leading coefficient: 1.2. Multiply 3 (the zero) by 1 and write the result under the next coefficient (-7): 3.3. Add -7 and 3 to get -4.4. Multiply 3 by -4 and write the result under the next coefficient (13): -12.5. Add 13 and -12 to get 1.6. Multiply 3 by 1 and write the result under the next coefficient (-3): 3.7. Add -3 and 3 to get 0 (as expected, since 3 is a zero).The result is: \(x^2 - 4x + 1\).

Step 4: Solve Quadratic Equation

The quotient from synthetic division is \(x^2 - 4x + 1\). To find its zeros, solve the quadratic equation \(x^2 - 4x + 1 = 0\) using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]where \(a = 1\), \(b = -4\), \(c = 1\).

Step 5: Apply Quadratic Formula

Substitute the values into the quadratic formula:\[x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(1)}}{2(1)} = \frac{4 \pm \sqrt{16 - 4}}{2}\]\[x = \frac{4 \pm \sqrt{12}}{2}\]\[x = \frac{4 \pm 2\sqrt{3}}{2}\]Simplifying gives: \(x = 2 \pm \sqrt{3}\). These are the other two zeros.

Key Concepts

Synthetic Division

Synthetic division is a simplified form of dividing polynomials, particularly when dividing by a linear factor of the form \((x - c)\), where \(c\) is a constant. It's highly efficient because it reduces the complexity and time required for polynomial division. Here's how it works step-by-step:

• Write down the zero at the top left of your setup. In our original exercise, this is 3, since it is a zero of the polynomial \(P(x)\).
• List the coefficients of the polynomial in descending order of their degrees. For \(x^3 - 7x^2 + 13x - 3\), that's 1, -7, 13, and -3.
• Bring down the leading coefficient (1) to the bottom row. This starts your result line.
• Multiply the zero by this leading coefficient and write it under the next coefficient. Add these two numbers to fill in the next spot in the bottom row.
• Repeat the multiply-and-add process until you've worked through all coefficients.

In the end, you'll have a new polynomial, which is the quotient, and a remainder. Here, the remainder is zero, confirming \((x - 3)\) is a factor. The resulting quotient is \(x^2 - 4x + 1\).

Quadratic Formula

The quadratic formula is essential for finding the roots of any quadratic equation. A quadratic equation is often represented as \(ax^2 + bx + c = 0\). The formula used is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here's how it helps:

• Variables: In the formula, \(a\), \(b\), and \(c\) are coefficients from your equation. In our quotient \(x^2 - 4x + 1\), \(a = 1\), \(b = -4\), and \(c = 1\).
• Discriminant: \(b^2 - 4ac\) is the discriminant. It determines the nature of the roots (real or complex).
• Substitution: Substitute your known values of \(a\), \(b\), and \(c\) into the formula to calculate the roots.

For our exercise, applying the formula to \(x^2 - 4x + 1\) gives the zeros \(x = 2 \pm \sqrt{3}\). These are the other solutions to the polynomial equation.

Finding Zeros

Finding zeros of polynomials means locating its roots - the values of \(x\) that make the polynomial equal to zero. These zeros give insights into the polynomial's graph and its factors.

• Known Zero: Sometimes, you are given a zero, allowing you to simplify the polynomial using division.
• Factoring: After division, you're left with a simpler polynomial. For example, from \(P(x) = x^3 - 7x^2 + 13x - 3\), synthetic division simplifies it to \(x^2 - 4x + 1\).
• Quadratic Solutions: Quadratic equations use the quadratic formula to find zeros; these zeros, along with any given, form the complete solution set for the polynomial equation.

Understanding zero properties helps solve polynomial equations more efficiently and reveals the behavior of polynomial graphs.

Polynomial Equations

Polynomial equations consist of variables raised to various powers, combined by multiplication and addition. They can vary in degree:

• Linear: First degree (e.g., \(x + 2\))
• Quadratic: Second degree (e.g., \(x^2 - 4x + 1\))
• Cubic and Beyond: Third degree and higher (e.g., \(x^3 - 7x^2 + 13x - 3\))

The key to solving them often involves:

• Factoring: Breaking down the equation into simpler parts.
• Expansion: Using given zeros to find factors.
• Solving Techniques: Methods like synthetic or long division for higher degrees and quadratic formula for quadratics.

Polynomial equations describe a vast range of scenarios, making them fundamental in mathematics. Understanding their solutions involves recognizing patterns and applying the right techniques for factoring and solving.