Problem 26 - Algebra for College Students

Question

Perform the indicated divisions. $$ \left(x^{4}+2 x^{3}-16 x^{2}+x+6\right) \div(x-3) $$

Step-by-Step Solution

Answer

The quotient is \(x^3 + 5x^2 - x - 2\) with a remainder of 0.

Step 1: Set Up the Division

Write down the polynomial division in long division format. Place \(x^4 + 2x^3 - 16x^2 + x + 6\) inside the division symbol and \(x - 3\) outside.

Step 2: Divide the First Term

Divide the leading term of the dividend \(x^4\) by the leading term of the divisor \(x\), which results in \(x^3\). Write \(x^3\) above the division bar.

Step 3: Multiply and Subtract

Multiply \(x^3\) by the divisor \(x - 3\) to get \(x^4 - 3x^3\). Subtract \(x^4 - 3x^3\) from the current dividend, \(x^4 + 2x^3\), resulting in \(5x^3\). Bring down the next term, \(-16x^2\), to get \(5x^3 - 16x^2\).

Step 4: Divide the Next Term

Divide \(5x^3\) by the leading term \(x\) of the divisor to get \(5x^2\). Write \(5x^2\) above the division bar next to \(x^3\).

Step 5: Multiply and Subtract Again

Multiply \(5x^2\) by the divisor \(x - 3\) to get \(5x^3 - 15x^2\). Subtract \(5x^3 - 15x^2\) from \(5x^3 - 16x^2\), resulting in \(-x^2\). Bring down the next term, \(x\), to form \(-x^2 + x\).

Step 6: Divide the Next Term

Divide \(-x^2\) by \(x\) to get \(-x\). Write \(-x\) above the division bar beside \(5x^2\).

Step 7: Multiply and Subtract Again

Multiply \(-x\) by \(x - 3\), resulting in \(-x^2 + 3x\). Subtract this from \(-x^2 + x\), which results in \(-2x\). Bring down the last term, \(6\), to get \(-2x + 6\).

Step 8: Divide the Last Term

Divide \(-2x\) by \(x\) to get \(-2\). Write \(-2\) above the division bar next to \(-x\).

Step 9: Final Multiply and Subtract

Multiply \(-2\) by \(x - 3\) to get \(-2x + 6\). Subtract this from \(-2x + 6\), resulting in a remainder of 0.

Key Concepts

Long Division

Polynomial long division is a technique similar to the long division process used with numbers, but it involves polynomials instead. This process is used to divide a polynomial by another polynomial of lower or equal degree, yielding a quotient and possibly a remainder.

Let's look at our example:

• We need to divide the polynomial \(x^4 + 2x^3 - 16x^2 + x + 6\) by \(x - 3\).
• The first step is to set up your long division by writing the divisor, \(x - 3\), to the left of the division symbol and the dividend, \(x^4 + 2x^3 - 16x^2 + x + 6\), under the division bar.

The division process involves: 1. **Dividing the leading term of the dividend** by the leading term of the divisor.2. **Multiplying and subtracting**: Multiply the result by the entire divisor, write below the dividend, then subtract from the dividend.3. **Bringing down the next term** of the polynomial from the dividend to continue the division.Repeat these steps until reaching the last term of the polynomial. If there are any terms remaining that can't be divided further, these would form the remainder.

Synthetic Division

Synthetic division is a simplified form of polynomial division, specifically used when dividing by linear polynomials of the form \(x - c\). It provides a quick and efficient method that eliminates the repetitive subtraction steps of long division. Though we used long division in the provided solution, synthetic division would dramatically speed up the process if the divisor is indeed linear:

• Firstly, identify \(c\) from the divisor \(x - 3\), which is \(3\).
• Write down the coefficients of the polynomial \(1, 2, -16, 1, 6\).
• Use synthetic setup by drawing a horizontal and vertical line to separate \(c\) from the list of coefficients.

The synthetic division comprises: 1. **Bring down the first coefficient**.2. **Multiply** this coefficient by \(c\) and add the result to the next coefficient.3. **Repeat** starting from the new result at each step.Ultimately, synthetic division concludes with the last number being the remainder. It's an excellent choice when you deal with linear divisors, offering a straightforward means to determine both quotient and remainder.

Polynomial Remainders

In polynomial division, the remainder is the portion of the dividend that is not fully divisible by the divisor. In our example, the division of \(x^4 + 2x^3 - 16x^2 + x + 6\) by \(x - 3\) resulted in a remainder of 0. This outcome means the divisor \(x - 3\) evenly divides the polynomial without any leftover terms, indicating \(x - 3\) is a factor of the polynomial.

The remainder theorem tells us that for a polynomial \(f(x)\) divided by \(x - c\), the remainder is \(f(c)\). In this case:

• Substituting \(x = 3\) in the polynomial \(f(x) = x^4 + 2x^3 - 16x^2 + x + 6\), confirms the remainder to be 0, showing \(x - 3\) is indeed a factor.
• If the remainder had been different from 0, that value itself would be \(f(3)\).

Understanding remainders in polynomial division helps identify factors and offers an insight into the divisibility of the polynomial, making it a pivotal tool in solving polynomial equations and simplifying expressions.