Problem 60 - Algebra for College Students

Question

The remainder and factor theorems are true for any complex value of \(c\). Therefore, for Problems \(56-58\), find \(f(c)\) by (a) using synthetic division and the remainder theorem, and (b) evaluating \(f(c)\) directly. $$ \text { Show that } x+3 i \text { is a factor of } f(x)=x^{4}+14 x^{2}+45 \text {. } $$

Step-by-Step Solution

Answer

Yes, \( x+3i \) is a factor of \( f(x) \).

Step 1: Identify function and divisor

First, identify the polynomial function \( f(x) = x^4 + 14x^2 + 45 \) and the divisor \( c = -3i \). We need to check if \( x + 3i \) is a factor of \( f(x) \), which is equivalent to checking if \( f(-3i) = 0 \).

Step 2: Apply synthetic division

Perform synthetic division of \( f(x) \) by \( x + 3i \). Use \( c = -3i \) for the synthetic division setup. Start with the coefficients of \( f(x) \), which are \( 1, 0, 14, 0, 45 \), where the missing \( x^3 \) and \( x^1 \) terms are represented with zeroes.

Step 3: Perform synthetic division computation

With \( -3i \) as the divisor:1. Bring down the leading coefficient: \( 1 \).2. Multiply \( 1 \) by \( -3i \), add to the next coefficient: result is \( 0 \).3. Multiply result by \( -3i \), add to the next coefficient: \( 14 \).4. Multiply result by \( -3i \), add to the next coefficient: \( 0 \).5. Multiply result by \( -3i \), add to the final coefficient \( 45 \).The remainder is computed to be \( 0 \).

Step 4: Remainder theorem conclusion

According to the remainder theorem, if the remainder is zero, then \( x+3i \) is a factor of \( f(x) \). Since the remainder from synthetic division is \( 0 \), this confirms that \( x+3i \) is indeed a factor.

Step 5: Evaluate \( f(c) \) directly

Substitute \( c = -3i \) into the polynomial:\[ f(-3i) = (-3i)^4 + 14(-3i)^2 + 45. \]Calculate each term:- \((-3i)^4 = 81\) because \((-3i)^4 = ((-3i)^2)^2 = (9)^2 = 81\).- \(14(-3i)^2 = 14(-9) = -126\).- So, \( f(-3i) = 81 - 126 + 45 = 0 \).

Step 6: Direct evaluation conclusion

The direct evaluation confirms that \( f(-3i) = 0 \). This also implies that \( x+3i \) is a factor of \( f(x) \), consistent with our earlier synthetic division result.

Key Concepts

Synthetic Division Explained

Synthetic Division is a shortcut method for dividing a polynomial by a binomial of the form \( x - c \). It is a more efficient alternative to long division for this purpose, and it's particularly useful when dealing with polynomials of higher degrees.
To perform Synthetic Division, follow these steps:

• Identify the coefficients of the polynomial. It's crucial to include placeholders (zeroes) for any missing terms.
• Pick the value of \( c \) from the divisor \( x - c \) or \( x + 3i \) as in our problem.
• Write the coefficients in a row and bring the first coefficient down as is.
• Multiply the down number by \( c \) and add the result to the next coefficient.
• Repeat this multiplication and addition process for each coefficient.
• The final number obtained is the remainder from the division.

By following these steps with the polynomial function \( f(x) = x^4 + 14x^2 + 45 \) and the divisor \( c = -3i \), we found that the remainder is zero, confirming the factor relationship.

Remainder Theorem Basics

The Remainder Theorem is a straightforward yet powerful concept in polynomial mathematics. It states that for any polynomial \( f(x) \) divided by \( x - c \), the remainder of this division is \( f(c) \).
This tells us something crucial: To know if \( x - c \) is a factor of \( f(x) \), simply evaluate \( f(c) \). If \( f(c) = 0 \), then \( x - c \) is a factor of \( f(x) \); otherwise, it is not. Remember, a factor leaves no remainder!
In our problem, we evaluate \( f(c) \) for the value \( c = -3i \):

• Plug \( -3i \) into the polynomial and perform the necessary calculations.
• If the result is zero, \( x + 3i \) is confirmed to be a factor.

Through direct calculation, we established that \( f(-3i) = 0 \). Thus, \( x + 3i \) truly divides the polynomial \( f(x) \) without leaving a remainder, proving the Remainder Theorem.

Understanding Complex Numbers

Complex Numbers are a type of number that includes the regular real number, as well as an imaginary unit \( i \), where \( i^2 = -1 \). They are typically expressed in the form \( a + bi \) where \( a \) and \( b \) are real numbers.
Complex Numbers allow us to solve polynomial equations that do not have solutions in the realm of Real numbers alone. This is essential for ensuring that every polynomial equation has a solution, known as the Fundamental Theorem of Algebra.
In our exercise, \( x + 3i \) is used to determine if it's a factor of the polynomial \( f(x) \). Complex arithmetic involves:

• Adding and subtracting complex numbers by combining their real and imaginary components separately.
• Multiplying involves using the distributive property, remembering \( i^2 = -1 \), and simplifying.

Understanding these operations is crucial for accurately performing calculations like in Step 5, where \( c = -3i \) is directly substituted into the polynomial to verify that \( x + 3i \) is indeed a factor.