Problem 19 - College Algebra with Corequisite Support

Question

For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor. $$ x^{3}+3 x^{2}+4 x+12 ; \quad x+3 $$

Step-by-Step Solution

Answer

The only real zero is \(x = -3\).

Step 1: Recap of the Factor Theorem

The Factor Theorem states that if \((x - c)\) is a factor of a polynomial \(f(x)\), then \(f(c) = 0\). This means that \(c\) is a root of the polynomial. In this problem, we are given \(x+3\) as a factor, which implies \(x = -3\) is a root of the polynomial \(f(x) = x^3 + 3x^2 + 4x + 12\).

Step 2: Verify the Given Factor

Using the given factor \(x + 3\), substitute \(x = -3\) into the polynomial: \(f(-3) = (-3)^3 + 3(-3)^2 + 4(-3) + 12 = -27 + 27 - 12 + 12 = 0\). Since \(f(-3) = 0\), \(x = -3\) is indeed a root, confirming \(x+3\) is a factor.

Step 3: Polynomial Division

To find the remaining factors, perform polynomial division of \(x^3 + 3x^2 + 4x + 12\) by \(x+3\). By dividing, we get: \((x^3 + 3x^2 + 4x + 12) \div (x+3) = x^2 + 0x + 4\). The quotient is \(x^2 + 4\).

Step 4: Solve for Remaining Zeros

Now, solve the quotient \(x^2 + 4 = 0\) for any additional real roots. Rearranging gives \(x^2 = -4\). Taking the square root on both sides, we find \(x = \pm \sqrt{-4}\). Since \(\sqrt{-4}\) involves imaginary numbers, there are no additional real zeros from this factor.

Step 5: Conclude Real Zeros

The only real zero of the polynomial is \(x = -3\), as the factor \(x^2 + 4\) contributes only non-real solutions.

Key Concepts

Factor Theorem

The Factor Theorem is a valuable tool in algebra for discovering the roots, or zeros, of a polynomial. It tells us that if

• \((x - c)\) is a factor of a polynomial \(f(x)\),
• then \(f(c) = 0\).

This means that \(c\) is a root of the polynomial. In simple terms, a root is a value of \(x\) that makes the polynomial equal to zero. If you can find a value \(c\) that satisfies \(f(c) = 0\), it confirms that \((x - c)\) is indeed a factor. In the given example, because \(x+3\) is a factor, the theorem tells us that \(x = -3\) is a root of our polynomial \(x^3 + 3x^2 + 4x + 12\). This provides a reliable starting point for further exploration of the polynomial’s properties.

Polynomial Division

Polynomial division allows us to simplify complex polynomials by dividing them and uncovering more factors or roots. It works similarly to long division with numbers, except we work with variables and coefficients. The steps are simple:

• Start with dividing the first term of your polynomial by the first term of the divisor.
• Multiply the entire divisor by the result and subtract that from the polynomial.
• Repeat the process with the new polynomial you get after subtraction.

In our exercise, we performed polynomial division of \(x^3 + 3x^2 + 4x + 12\) by the given factor \(x+3\). The division yielded \(x^2 + 4\) as the quotient. This means that the original polynomial can be expressed as the product of \((x+3)(x^2+4)\). Understanding this process helps in breaking down the problem into easier, more manageable parts, allowing us to find other potential roots or factors within the remaining polynomial expression.

Real Zeros

Real zeros, or real roots, of a polynomial are the values of \(x\) that satisfy the polynomial equation as zero. Identifying these values is essential both in theoretical mathematics and practical applications.In the context of our exercise, once we identified \(x = -3\) as a real root using the Factor Theorem, the next step was to explore other potential roots from the quotient \(x^2+4\). Solving the equation \(x^2 + 4 = 0\) involves rearranging it to \(x^2 = -4\), followed by solving for \(x\). This leads to \(x = \pm \sqrt{-4}\). Since the square root of a negative number involves imaginary numbers, \(x^2 + 4\) doesn't contribute additional real roots.Thus, the only real zero we find is \(x = -3\). The understanding of real zeros is significant, as these values often represent important points such as intercepts in graphs or solutions to real-world problems where the polynomial models a phenomenon.