Problem 27 - Precalculus with Calculus Previews

Question

Use synthetic division to find the quotient \(q(x)\) and remainder \(r\) when \(f(x)\) is divided by the given linear polynomial. $$ f(x)=x^{4}+16 ; x-2 $$

Step-by-Step Solution

Answer

The quotient is \(x^3 + 2x^2 + 4x + 8\) and the remainder is \(32\).

Step 1: Setup the Synthetic Division Table

Identify the coefficients of the polynomial \(f(x) = x^4 + 0x^3 + 0x^2 + 0x + 16\). Since there are missing terms of \(x^3, x^2, \) and \(x\), include these zero coefficients. This gives the coefficients \([1, 0, 0, 0, 16]\).The divisor is \(x - 2\), so the value to use for synthetic division is \(2\). Write these numbers in a row with \(2\) to the left.

Step 2: Begin Synthetic Division

Bring down the leading coefficient \(1\) from the setup row to the bottom row.Start the process by multiplying the value on the bottom row (initially \(1\)) by \(2\), placing the product under the next coefficient.

Step 3: Continue the Division Process

Add each column in the setup row to the product written below it. Record each sum in the bottom row.1. Multiply \(1\) (bottom row) by \(2\) (outside the table). Write \(2\) underneath \(0\).2. Add \(0 + 2 = 2\). Write \(2\) in the bottom row.3. Multiply \(2\) (bottom row) by \(2\). Write \(4\) beneath the next \(0\).4. Add \(0 + 4 = 4\). Write \(4\) in the bottom row.5. Multiply \(4\) (bottom row) by \(2\). Write \(8\) beneath the next \(0\).6. Add \(0 + 8 = 8\). Write \(8\) in the bottom row.7. Multiply \(8\) (bottom row) by \(2\). Write \(16\) beneath \(16\).8. Add \(16 + 16 = 32\). Write \(32\) in the bottom row.

Step 4: Identify the Quotient and Remainder

The bottom row after addition, \([1, 2, 4, 8, 32]\), represents the coefficients of the resulting polynomial and the remainder.Thus, the quotient \(q(x)\) is \(x^3 + 2x^2 + 4x + 8\), and the remainder is \(32\).

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another. Think of it like the long division you learned in elementary school but with polynomials. In our problem, synthetic division is used to divide the polynomial function, \( f(x) = x^4 + 16 \), by the linear polynomial \( x - 2 \).

The process involves simplifying the division by focusing only on the coefficients of the polynomials involved. This is where synthetic division becomes handy, as it skips writing out all the x terms explicitly and deals only with the numbers, speeding up the calculation process. Instead of writing out variables and powers, synthetic division lets us simplify using a table or process, making polynomial division easier to manage, especially with higher-degree polynomials.

By understanding polynomial division, we recognize how larger expressions can be broken down into simpler, related components, just like breaking down numbers in division and seeing how they relate.

Remainder Theorem

The Remainder Theorem is a quick way to find the remainder of a polynomial division. It states that if a polynomial \( f(x) \) is divided by a linear divisor \( x - c \), the remainder of this division is \( f(c) \).

In simpler words, if you substitute the value \( c \) into the polynomial, the result will directly give you the remainder. For example, with our given polynomial \( f(x) = x^4 + 16 \) and divisor \( x - 2 \), substituting \( x = 2 \) into \( f(x) \) gives us the remainder.

This theorem provides a powerful way to check polynomial division results, as it helps verify the correct remainder without repeating the division steps all over again.

Quotient of a Polynomial

The quotient in polynomial division is what you get when a polynomial, like \( f(x) \), is divided by another polynomial, such as \( x - 2 \). The process of finding this quotient involves performing synthetic division or traditional long division on the coefficients.

For our polynomial \( f(x) = x^4 + 16 \), the synthetic division showed a resulting quotient of \( q(x) = x^3 + 2x^2 + 4x + 8 \). This means that when \( x^4 + 16 \) is divided by \( x - 2 \), it fits the pattern described in the quotient without leaving any terms unchecked, except for the final remainder.

Understanding the quotient helps us see how one polynomial can be broken down and compared to another divisively, showing the relationship between the original polynomial and its divisor.

Factor Theorem

The Factor Theorem is closely related to the Remainder Theorem and has applications in determining if a certain linear polynomial, \( x - c \), is a factor of another polynomial. According to this theorem, if \( f(c) = 0 \) for a polynomial \( f(x) \), then \( x - c \) is a factor of \( f(x) \).

In our context, with \( f(x) = x^4 + 16 \), if substituting \( x = c \) resulted in a remainder of zero during division, it would imply that \( x - 2 \) is a factor of the polynomial. However, the remainder was 32, not zero, indicating that \( x - 2 \) is not a factor.

Thus, the Factor Theorem helps in determining which divisors are true factors of a polynomial, allowing for further simplification and understanding of polynomial expressions.