Problem 24 - College Algebra

Question

Divide using synthetic division. $$ \left(x^{5}+4 x^{4}-3 x^{2}+2 x+3\right) \div(x-3) $$

Step-by-Step Solution

Answer

The result of the division is \(x^{4} + 7x^{3} + 21x^{2} + 60x + 182 + \frac{539}{x-3}\).

Step 1: Write Down the Coefficients

Write down the coefficients of the terms of the polynomial in descending order, including a 0 for any missing terms. So for \(x^{5}+4 x^{4}-3 x^{2}+2 x+3\), the coefficients will be [1, 4, 0, -3, 2, 3].

Step 2: Write Down the Divisor and Set Up the Synthetic Division Table

Write down the value of 'a' in 'x-a'. Here, 'a' is 3 as we are dividing by 'x - 3'. Place the 'a' value to the left of a vertical line and the coefficients to the right. 3 | 1 4 0 -3 2 3 --

Step 3: Begin Synthetic Division

Drop the first coefficient down, as is, to the row below. Then, multiply the 'a' value by this resulting value and write the answer underneath the second coefficient. Add the second coefficient and this new number:3 | 1 4 0 -3 2 3 | 3 -------------- | 1 7 Continue this process across the whole set of coefficients.

Step 4: Complete Synthetic Division

The final row of numbers represents the coefficients of the answer to the division:3 | 1 4 0 -3 2 3 | 3 21 63 180 536 -------------- | 1 7 21 60 182 539The first number is the coefficient of the \(x^4\) term, the second number is the coefficient of the \(x^3\) term, and so forth. The final number is the remainder. Written out, the result is: \(x^{4} + 7x^{3} + 21x^{2} + 60x + 182 + \frac{539}{x-3}\)

Key Concepts

Polynomial Long Division

Think of polynomial long division as the algebraic version of long division you learned with numbers, but this time with variables included. It's a structured process used to divide a degree of one polynomial by another. The method is similar to numerical long division: You divide the highest degree terms, multiply the divisor by the result, subtract, bring down the next term, and repeat until all terms are used. This helps break down complex polynomial expressions into more manageable pieces.

Steps to Follow

• Divide the first term of the dividend by the first term of the divisor.
• Multiply the divisor by the result and subtract it from the dividend.
• Bring down the next term and repeat the process.
• Continue until there are no terms left to bring down.

While polynomial long division is thorough, it can be quite lengthy, especially with higher-degree polynomials. It becomes critical to ensure each step is followed accurately to arrive at the correct quotient and remainder.

Division of Polynomials

Division of polynomials involves finding the quotient when one polynomial is divided by another. The key is recognizing that polynomials are algebraic expressions that consist of variables raised to whole-number exponents and their coefficients. There are different ways to divide polynomials, but the most common are polynomial long division and synthetic division, each with its own use cases.

Synthetic division is a shortcut method that simplifies the process and is typically used when dividing by a linear term. It is faster and usually easier once you understand the process, which involves using only the coefficients of the polynomials.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. Polynomials are a type of algebraic expression that include terms which are variables (like 'x') raised to whole-number exponents, multiplied by coefficients. These expressions can have one term (monomial), two terms (binomial), or more (trinomial, etc.).

Working with algebraic expressions requires understanding how to manipulate them using algebraic rules. This includes being able to add, subtract, multiply, and divide expressions, which often involves factoring, expanding, and simplifying terms. Mastering algebraic expressions is fundamental to solving more complex equations and understanding advanced mathematical concepts.

Remainder Theorem

The Remainder Theorem is a fascinating and handy tool in polynomial algebra. It states that when a polynomial, say, 'f(x)', is divided by a linear divisor of the form 'x - a', the remainder of the division is the value of the polynomial evaluated at 'a', which is 'f(a)'.

For example, if you divide a polynomial by 'x - 3', the remainder of this division equals the value of the polynomial when 'x' is 3. This theorem not only provides insights about the remainder without doing long division but also helps in checking the correctness of the division if you've already found it.