Problem 35 - College Algebra

Question

For the following exercises, use synthetic division to find the quotient. $$ \left(x^{4}-12 x^{3}+54 x^{2}-108 x+81\right) \div(x-3) $$

Step-by-Step Solution

Answer

The quotient is \(x^3 - 9x^2 + 27x - 27\).

Step 1: Set up the Synthetic Division

First, note the divisor is \(x-3\), so we use \(3\) in synthetic division. Write down the coefficients of the dividend, \(x^4 - 12x^3 + 54x^2 - 108x + 81\), which are \(1, -12, 54, -108, 81\). Prepare a row for these coefficients.

Step 2: Bring Down the Leading Coefficient

Write the first coefficient, \(1\), below the line to begin the process.

Step 3: Multiply and Add

Multiply \(3\) (the number from \(x-3\)) by the first coefficient below the line \(1\). So, \(3 \times 1 = 3\). Write \(3\) under the next coefficient, \(-12\). Add \(-12 + 3 = -9\). Write \(-9\) below the line.

Step 4: Repeat Multiply and Add

Multiply \(3\) by the result just written, \(-9\). Get \(3 \times -9 = -27\). Write \(-27\) under the next coefficient, \(54\). Add \(54 + (-27) = 27\). Write \(27\) below the line.

Step 5: Continue the Process

Multiply \(3\) by \(27\). Calculate \(3 \times 27 = 81\). Write \(81\) under the next coefficient, \(-108\). Add \(-108 + 81 = -27\). Write \(-27\) below the line.

Step 6: Final Multiplication and Addition

Multiply \(3\) by \(-27\). Calculate \(3 \times -27 = -81\). Write \(-81\) under the last coefficient, \(81\). Add \(81 + (-81) = 0\). The remainder is \(0\), confirming the division is exact.

Step 7: Write the Quotient

The result below the line, \(1, -9, 27, -27\), are the coefficients of the quotient polynomial: \(x^3 - 9x^2 + 27x - 27\).

Key Concepts

Polynomial Division

Polynomial division, similar to long division with numbers, is a method used to divide one polynomial by another. It is particularly useful when dealing with higher-degree polynomials. In synthetic division, a simplified form of polynomial division, we focus on cases where the divisor is a linear binomial of the form \(x - c\). This method relies heavily on manipulating the coefficients of the polynomials without writing out variable terms.

Let's consider the division of a polynomial \(x^4 - 12x^3 + 54x^2 - 108x + 81\) by a binomial \(x - 3\). The aim is to find another polynomial, called the quotient, and possibly a remainder; however, in this exercise, there is no remainder as it divides perfectly.

• Identify the coefficients of the dividend polynomial.
• Set up the division using these coefficients and the zero of the divisor \(x - 3\).
• Perform repetitive multiplication and addition steps.

Remainder Theorem

The Remainder Theorem is a convenient rule when working with polynomial division. This theorem states that if you divide a polynomial \(f(x)\) by a linear divisor \(x-c\), the remainder of this division is simply \(f(c)\). In other words, if \(f(x)\) is divided by \(x-3\), evaluate \(f(3)\) to find the remainder.

In our synthetic division example:

• We found that \(f(3) = 81 - 108 + 54 - 12 + 1 \) simplifies to zero.
• This confirms the remainder is zero, validating our division results.

This theorem is particularly useful for quickly checking polynomial division results without going through the entire division process.

Factor Theorem

The Factor Theorem is a special case of the Remainder Theorem, providing a direct connection between division and factoring polynomials. It implies that if \(x - c\) is a divisor of \(f(x)\) and the remainder is zero, then \(x - c\) is a factor of \(f(x)\). This means the polynomial can be rewritten as a product of \(x - c\) and the quotient polynomial found during division.

In our exercise:

• The remainder is zero confirming that \(x - 3\) is indeed a factor of the polynomial \(x^4 - 12x^3 + 54x^2 - 108x + 81\).
• This allows us to express the polynomial in a factored form using the quotient polynomial obtained.

Applying the Factor Theorem helps in simplifying complex polynomials and solving polynomial equations.

Quotient Polynomial

The quotient polynomial is the polynomial obtained as a result of division, excluding the remainder. The degree of this polynomial is one less than that of the original polynomial. In the context of synthetic division, the process works through coefficients efficiently, yielding the quotient polynomial.

From our exercise:

• After synthetic division of \(x^4 - 12x^3 + 54x^2 - 108x + 81\) by \(x - 3\), the coefficients \(1, -9, 27, -27\) form the new polynomial.
• This quotient polynomial is \(x^3 - 9x^2 + 27x - 27\).
• Thus, the quotient polynomial represents a simpler form or factor of the original polynomial.

Understanding quotient polynomials is crucial, especially in solving higher degree polynomial equations efficiently through division techniques.