Problem 62 - A Graphical Approach to College Algebra

Question

For each polynomial, at least one zero is given. Find all others analytically. $$P(x)=x^{3}+2 x^{2}-11 x-12 ; \quad 3$$

Step-by-Step Solution

Answer

The zeros of the polynomial are 3, -4, and -1.

Step 1: Verify Given Zero

We are given that 3 is a zero of the polynomial \(P(x)=x^{3}+2x^{2}-11x-12\). We can confirm this by substituting \(x=3\) into the polynomial:\[\begin{align*} P(3) &= 3^3 + 2 \cdot 3^2 - 11 \cdot 3 - 12 \ &= 27 + 18 - 33 - 12 \ &= 0. \end{align*}\] Since \(P(3) = 0\), 3 is indeed a zero of the polynomial.

Step 2: Perform Polynomial Division

Use synthetic division to divide \(P(x)\) by \(x-3\), since \(3\) is a root. Write the coefficients: \(1, 2, -11, -12\). Perform the synthetic division process:1. Bring down the first coefficient: 1.2. Multiply by 3 (root), add to next coefficient: - \(1 \cdot 3 = 3\); \(2 + 3 = 5\)3. Multiply 5 by 3, add to next coefficient: - \(5 \cdot 3 = 15\); \(-11 + 15 = 4\)4. Multiply 4 by 3, add to next coefficient: - \(4 \cdot 3 = 12\); \(-12 + 12 = 0\)The quotient is \(x^2 + 5x + 4\). The remainder is zero, confirming the division.

Step 3: Find Zeros of the Quotient Polynomial

Determine the zeros of the quadratic polynomial \(x^2 + 5x + 4\) by factoring. The polynomial factors as \((x + 4)(x + 1)\). Thus, the zeros of this polynomial are the solutions to:\[x + 4 = 0 \quad \text{or} \quad x + 1 = 0\]Solving these equations gives the additional zeros: \(x = -4\) and \(x = -1\).

Step 4: Compile All Zeros

List all the zeros of \(P(x)\): The given zero is \(3\), and the zeros we found from the quadratic are \(-4\) and \(-1\). Therefore, the complete set of zeros of the polynomial are \(x = 3, -4, -1\).

Key Concepts

Synthetic Division

Synthetic Division is a simple technique to divide a polynomial by a binomial of the form \(x - c\). It's an efficient and less cumbersome method than traditional long division. Imagine needing a quick shortcut to find out what's left when dividing; that's what synthetic division is for!
Here's how you perform it:

• Write down the coefficients of the polynomial in order. For our exercise, these are 1, 2, -11, -12.
• Use the root you know, in this case, 3, and position it at the side.

Then, follow these steps:1. **Bring down** the first coefficient (1) straight down.2. Multiply this coefficient by the root (3) and **add** the result to the next coefficient (2). Repeat this until you've processed all coefficients.
As you multiply and add, you'll notice a pattern that makes it quicker than traditional ways. This ease comes from dealing only with numbers, eliminating variables until the end.Finally, you find what's left: for our polynomial, the quotient is \(x^2 + 5x + 4\). The remainder provides a check; if it is 0, your division is exact.

Factoring Quadratics

Factoring quadratics, like \(x^2 + 5x + 4\), is about breaking down a quadratic expression into simpler, multiplied components. This helps reveal the zeros or solutions of the equation.
To factor effectively:

• Look for two numbers that multiply to produce the constant term (4) and add to give the middle coefficient (5).
• These numbers are 4 and 1 for our equation, since 4 + 1 = 5 and 4 * 1 = 4.

Write the quadratic as \((x + 4)(x + 1)\). When you expand this, it returns to \(x^2 + 5x + 4\), verifying that the factorization is correct. Factoring reveals that the zeros of this quadratic, where the graph intersects the x-axis, are \(x = -4\) and \(x = -1\). Factoring is a powerful method since it simplifies solving quadratics into basic arithmetic.

Polynomial Division

Polynomial Division, as a broader concept, involves dividing one polynomial by another, revealing useful information like factors or simplified forms.
We've seen Synthetic Division as a specific, pared-down case focusing on linear divisors, but Polynomial Division can involve more complex scenarios. Here's what it entails:

• Just like long division of numbers, divide the leading term of the dividend by the leading term of the divisor.
• Subtract the product from the dividend and repeat the process with the new polynomial formed.

This procedure continues until you can't divide any further, potentially leaving a polynomial remainder. The quotient provides insights into the structure of the polynomial. In our exercise, using Synthetic Division led us to divide \(x^3 + 2x^2 - 11x - 12\) by \(x - 3\), exemplifying Polynomial Division's practical use.

Verify Polynomial Zeros

Verifying Polynomial Zeros is the process of checking whether a suggested root actually satisfies the polynomial equation until it equals zero.
For our exercise, we confirmed that 3 is a zero by substituting it into \(P(x) = x^3 + 2x^2 - 11x - 12\). When plugged in, the calculations yielded zero, thus proving that it is indeed a valid zero.
This step is crucial as:

• It ensures the correctness of the root discovery process.
• Confirms that the division steps, whether synthetic or traditional, were performed accurately.

Always plug the zero back into the original polynomial. If the polynomial evaluates to zero, you’ve verified correctly. The result is a complete set of roots that solve the polynomial equation. For our polynomial, confirming the zero led us to reliably factor the remainder and find all other zeros: \(-4\) and \(-1\).