Problem 23 - College Algebra

Question

For the following exercises, use synthetic division to find the quotient. $$ \left(4 x^{3}-5 x^{2}+13\right) \div(x+4) $$

Step-by-Step Solution

Answer

The quotient is \(4x^2 - 21x + 84\) with a remainder of \(-323\).

Step 1: Understand Synthetic Division

Synthetic division is a shortcut method of dividing a polynomial by a binomial of the form \((x - c)\). Here, we want to divide \(4x^3 - 5x^2 + 0x + 13\) by \((x + 4)\), so \(c = -4\). Make sure to include a term for any missing powers of \(x\), like the \(0x\) term here.

Step 2: Set Up the Synthetic Division

Write the coefficients of the polynomial \(4, -5, 0, 13\) in a row. Write \(-4\) on the left side to represent the divisor \((x + 4)\) since we use \(c = -4\).

Step 3: Perform Synthetic Division

1. Bring down the first coefficient \(4\) as it is.2. Multiply \(-4\) by \(4\) and write \(-16\) under the next coefficient.3. Add \(-5\) and \(-16\) to get \(-21\).4. Multiply \(-4\) by \(-21\) and write \(84\) under the next coefficient.5. Add \(0\) and \(84\) to get \(84\).6. Multiply \(-4\) by \(84\) and write \(-336\) under the last coefficient.7. Add \(13\) and \(-336\) to get \(-323\).

Step 4: Write the Result

The numbers \(4, -21, 84\) are the coefficients of the quotient polynomial, and \(-323\) is the remainder. So the quotient is \(4x^2 - 21x + 84\), with a remainder of \(-323\). Thus, the division can be expressed as: \[4x^2 - 21x + 84 - \frac{323}{x+4}\].

Key Concepts

Polynomials

Polynomials are expressions that consist of variables and coefficients, involving operations such as addition, subtraction, and multiplication. They are arranged in terms of powers of the variables, typically written in descending order of the exponents. For example, in the exercise provided, we are working with the polynomial \(4x^3 - 5x^2 + 13\). This is a degree 3 polynomial because the highest exponent of \(x\) is 3. It has the following features:

• The terms are listed by descending powers of \(x\): \(4x^3, -5x^2, 0x, 13\).
• Each term of the polynomial consists of a coefficient (like 4 or -5) and may also involve a variable (like \(x\) or \(x^2\)).
• Polynomials can have one or more terms and are defined over a mathematical field, usually the field of real numbers.

Understanding polynomials is crucial as they form the basis for the division we perform in synthetic division.

Division Algorithm

The division algorithm for polynomials is analogous to long division with numbers. When dividing polynomials, the goal is to express the original polynomial as the product of the divisor and the quotient, plus a remainder. When we have a polynomial \(P(x)\) and we divide it by another polynomial \(D(x)\), according to the division algorithm:\[P(x) = D(x) \cdot Q(x) + R(x)\]where \(Q(x)\) is the quotient polynomial and \(R(x)\) is the remainder polynomial.
A few points to consider while using the division algorithm:

• The degree of the remainder \(R(x)\) must be less than the degree of the divisor \(D(x)\).
• The division algorithm helps in simplifying complex polynomial expressions and is used widely in calculus and algebra.

The division algorithm becomes more efficient with the use of synthetic division, especially when dividing by a binomial.

Remainder Theorem

The Remainder Theorem provides insight into polynomial division, specifically when a polynomial \(P(x)\) is divided by a linear divisor of the form \((x - c)\). According to the theorem, the remainder of this division is the value of the polynomial at \(c\), which is \(P(c)\). It's a useful shortcut because:

• It allows us to quickly determine the remainder without performing the full division.
• In the synthetic division example given, where \(x + 4\) is the divisor, we have \(c = -4\). The remainder in the division process was found to be \(-323\), meaning \(P(-4) = -323\).

Thus, the theorem seamlessly connects the algebraic manipulation of polynomials with evaluation, helping us validate our synthetic division results.

Binomial Division

Binomial division involves dividing a polynomial by a simple binomial, typically of the form \((x - c)\). Synthetic division is a streamlined method for performing binomial division where:\

• Only the coefficients of the polynomial are involved in the division process.
• The calculation is simplified significantly by avoiding direct manipulation of the variable expressions.
• In our example \(4x^3 - 5x^2 + 13\) divided by \(x + 4\) requires \(-4\) as the number used in synthetic division because \(x + 4\) can be rewritten as \((x - (-4))\).

By focusing on the coefficients, synthetic division provides an efficient means to obtain the quotient and remainder, enhancing computational speed and accuracy compared to traditional polynomial long division.