Problem 35 - College Algebra and Calculus: An Applied Approach

Question

Use synthetic division to divide. Divisor \(x+\frac{1}{2}\) Dividend $$4 x^{3}+16 x^{2}-23 x-15$$

Step-by-Step Solution

Answer

The quotient from the synthetic division of \(4x^3 + 16x^2 - 23x - 15\) by \(x + \frac{1}{2}\) is \(4x^2 + 14x -7\) with a remainder of -\frac{7}{2}.

Step 1: Prepare for Synthetic Division

Arrange the coefficients of the dividend: 4, 16, -23, -15 in a row. The divisor is \(x + \frac{1}{2}\), so -\frac{1}{2} will be used in the synthetic division process.

Step 2: Begin the Division Process

Bring down the first coefficient which is 4. Multiply -\frac{1}{2} (from the divisor) by 4 and write the result beneath the second coefficient in a column.

Step 3: Continue the Division Process

Add the numbers in the second column. Multiply the result by -\frac{1}{2} and write it under the third coefficient. Repeat the same process for all remaining coefficients in the dividend.

Step 4: Conclusion of Synthetic Division

At the end of the synthetic division process, the numbers in the bottom row are the coefficients of the quotient and the remainder. The degrees of the quotient terms are one degree less than the corresponding dividend terms.

Key Concepts

Polynomial Long Division

Polynomial long division is a method similar to traditional long division, but it is used to divide polynomials instead of integers. This technique is very useful in algebra when we need to simplify complex polynomial expressions or when dividing one polynomial by another.

To begin, first set up the division problem similar to how you would with numbers, by placing the dividend (the polynomial to be divided) inside the division bracket and the divisor (the polynomial you are dividing by) outside. Just like with numbers, you focus on dividing the leading terms, finding a quotient that will cancel out the first term when multiplied by the divisor, and place it above the division bar. This is repeated for each term in the dividend until no terms are left or the remainder is of smaller degree than the divisor.

Polynomial long division can be quite lengthy and requires careful attention to detail, but the process reinforces important algebraic concepts such as the distribution of multiplication over addition and how polynomial terms of the same degree can be compared and reduced.

Algebraic Division

Algebraic division encompasses methods like polynomial long division and synthetic division, tailored for simplifying or solving algebraic expressions. Through algebraic division, we deal with variables and unknowns, unlike simple arithmetic division which focuses on numbers.

The purpose of algebraic division is to break down complex algebraic expressions into simpler ones, which makes identifying zeros, factors, and solutions of polynomial equations manageable. When approaching algebraic division, students should remember the importance of manipulating variables while respecting the fundamental laws of algebra. Steps include aligning like terms, watching for negative signs and subtraction, and reducing expressions to their lowest forms. Through practice, recognizing patterns and familiarizing oneself with standard forms can make this process more intuitive.

Remainder Theorem

The remainder theorem is a key concept in algebra which tells us that when a polynomial, f(x), is divided by a linear divisor of the form (x - c), the remainder of that division is f(c). This theorem provides a shortcut for finding the remainder without performing the entire polynomial division.

Applying the remainder theorem is straightforward: after identifying the divisor (x - c), simply plug the value of c into the polynomial as if it were x. The resulting value is the remainder. For example, if you have a polynomial f(x) and you divide it by (x - 2), then the remainder of that division is the number you get when you evaluate f(2). This theorem becomes particularly useful when it can be determined that the polynomial has a remainder of zero, indicating that (x - c) is a factor of the polynomial.