Problem 25 - College Algebra

Question

For the following exercises, use synthetic division to find the quotient. $$ \left(x^{3}-21 x^{2}+147 x-343\right) \div(x-7) $$

Step-by-Step Solution

Answer

The quotient is \(x^2 - 14x + 49\).

Step 1: Set Up the Synthetic Division

Write down the coefficients of the dividend polynomial: 1, -21, 147, and -343. Then, write the zero of the divisor, which is 7, to the left of the coefficients.

Step 2: Bring Down the Leading Coefficient

Bring down the leading coefficient of the polynomial, 1, directly below the line.

Step 3: Multiply and Add

Multiply the divisor zero (7) by the number below the line (1) and write the result (7) underneath the second coefficient (-21). Add these two numbers: \(-21 + 7 = -14\). Write -14 below the line.

Step 4: Repeat Multiply and Add

Repeat the multiplication and addition with the new number below the line. Multiply 7 by -14 to get -98 and add it to the next coefficient (147) to get \(147 - 98 = 49\). Write 49 below the line.

Step 5: Final Multiply and Add

Multiply 49 by the divisor zero (7), giving 343. Add this to the final coefficient (-343): \(-343 + 343 = 0\). This zero confirms the divisor is a factor.

Step 6: Write the Quotient

The numbers below the line, 1, -14, and 49, represent the coefficients of the quotient polynomial. Thus, the quotient is \(x^2 - 14x + 49\).

Key Concepts

Polynomial Division

Polynomial division is an essential tool used for dividing one polynomial by another. It helps simplify complex expressions, allowing you to determine factors and simplify algebraic expressions. There are various techniques to perform polynomial division, with synthetic division being a common one for specific cases.

When performing polynomial division:

• The polynomial to be divided is called the dividend.
• The polynomial you are dividing by is called the divisor.
• The result of the division includes a quotient and possibly a remainder.

Synthetic division is a simplified form of polynomial division that works only when the divisor is a linear polynomial of the form \(x - c\). It uses the coefficients of the polynomial for straightforward calculation. In our case, the dividend is \(x^3 - 21x^2 + 147x - 343\) and the divisor is \(x - 7\). This method is powerful because it reduces the complexity of polynomial division to a series of multiplications and additions.

Quotient Polynomial

The quotient polynomial is the result you get when you successfully divide the dividend by the divisor in the context of polynomial division. In synthetic division, after performing the necessary arithmetic operations, the sequence of numbers you get at the bottom row (excluding the last) represents the coefficients of the quotient polynomial.

In the exercise given:

• After completing the synthetic division, we obtained the coefficients 1, -14, and 49.
• These coefficients correspond to the polynomial \(x^2 - 14x + 49\).

This polynomial \(x^2 - 14x + 49\) is our quotient. It tells us what happens when we divide the polynomial \(x^3 - 21x^2 + 147x - 343\) by \(x - 7\). The outcome shows that the original polynomial can be expressed as the product of \(x - 7\) and \(x^2 - 14x + 49\), demonstrating the divisor is a factor of the dividend.

Remainder Theorem

The remainder theorem states that when you divide a polynomial \(f(x)\) by a linear divisor \(x - c\), the remainder of this division is \(f(c)\). This theorem is a strong indicator that aids not only in finding roots but also in verifying factors of polynomials.

In the given problem, we applied synthetic division with the divisor \(x - 7\):
- The last number in the row after completing the synthetic division is the remainder. In this case, the remainder is 0.
- A remainder of 0 indicates that \(x - 7\) is indeed a factor of the polynomial \(x^3 - 21x^2 + 147x - 343\).

This makes sense because substituting 7 into the polynomial will yield zero, confirming the presence of 7 as a root according to the remainder theorem. It verifies that the polynomial can be exactly divided by the linear divisor without leaving any remainder.