Problem 26 - Algebra for College Students

Question

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(3 x^{4}-x^{3}+2 x^{2}-7 x-1\right) \div(x+1) $$

Step-by-Step Solution

Answer

Quotient: \(3x^3 - 4x^2 + 6x - 13\), Remainder: 12.

Step 1: Set Up the Synthetic Division

First, identify the coefficients of the dividend polynomial \(3x^4-x^3+2x^2-7x-1\). They are: 3, -1, 2, -7, and -1. For the divisor \(x+1\), set \(x+1=0\) to find the root \(-1\). Use this root for synthetic division.

Step 2: Perform Synthetic Division

Write the coefficients in a row: 3, -1, 2, -7, -1. Bring down the first coefficient, 3. Then, multiply it by the root (-1) and add to the next coefficient: \(3 \times (-1) = -3\); add to -1: \(-1 + (-3) = -4\). Repeat this: \(-4 \times (-1) = 4\), add to 2: \(2 + 4 = 6\). Continue these steps: \(6 \times (-1) = -6\), add to -7: \(-7 + (-6) = -13\). Finally, \(-13 \times (-1) = 13\), add to -1: \(-1 + 13 = 12\).

Step 3: Interpret the Result

The coefficients from synthetic division: 3, -4, 6, -13 with a remainder of 12. The quotient polynomial is one degree lower than the original, so it is: \(3x^3 - 4x^2 + 6x - 13\). The remainder is 12.

Step 4: Write the Final Answer

The result of \((3x^4-x^3+2x^2-7x-1) \div (x+1)\) is: quotient \(3x^3 - 4x^2 + 6x - 13\) and remainder 12.

Key Concepts

Polynomial Division

Polynomial division is a mathematical process similar to division with numbers, but it involves dividing polynomials. It is used to find how one polynomial can be split by another. Just like how we do long division with numbers, polynomial division might seem complex at first but becomes manageable once broken into steps.
This method helps us simplify polynomial expressions, solving equations and finding factors.
If the division results in a polynomial of a lower degree and a remainder, it means the first polynomial isn’t wholly divisible by the second.
For example, when dividing \(3x^4 - x^3 + 2x^2 - 7x - 1\) by \(x + 1\), synthetic division offers a straightforward way to work through this process by focusing on coefficients.

Quotient and Remainder

In polynomial division, the terms 'quotient' and 'remainder' are pivotal. Just like dividing numbers, when you divide polynomials the quotient is what remains after the division is done, and the remainder is what’s left over.
Using synthetic division simplifies this process, by allowing you to focus only on the coefficients and reduces the complexity associated with the variable terms.
For the example provided, the quotient found from the division \((3x^4 - x^3 + 2x^2 - 7x - 1) \div (x + 1)\) is \(3x^3 - 4x^2 + 6x - 13\), and the remainder is 12.
This means the original polynomial can be expressed as:

• Quotient Polynomial: \(3x^3 - 4x^2 + 6x - 13\)
• Remainder: 12

Such expressions can be vital in understanding the behavior of polynomial functions and are important in fields like calculus.

Roots of Polynomial Division

Roots play an essential role in synthetic division, as they help simplify the process. In the given problem, the divisor is \(x+1\). To carry out synthetic division, we need to find the root of this expression, which involves setting the expression equal to zero.
This gives us the equation \(x + 1 = 0\), leading to a root of \(x = -1\). This specific root is used to perform synthetic division on the coefficients of the polynomial.
Using the root, we adjust each coefficient in sequence to quickly find the quotient and remainder. This efficiently reveals how the polynomial behaves with the divisor, making it faster and less error-prone than other methods. Identifying and using roots is foundational to understanding how divisors interact with polynomials across different mathematical problems.