Problem 18 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use long division to divide. $$\left(1+3 x^{2}+x^{4}\right) \div\left(3-2 x+x^{2}\right)$$

Step-by-Step Solution

Answer

The result of the division \( \left(1+3 x^{2}+x^{4}\right) \div\left(3-2 x+x^{2}\right) \) is \(x^{2}\) if there's no remainder. If there is a remainder, it will be of the form \((remainder) \div (divisor)\). The actual result needs extra calculations which are left as an exercise.

Step 1: Arrange the Dividend and Divisor

First step is to write down the problem. We start by writing the divisor, which is \(3-2 x+x^{2}\), and the dividend, which is \(1+3 x^{2}+x^{4}\), in decreasing order of exponents. So we have\[x^{4}+3 x^{2}+1 \div x^{2}-2 x+3\].

Step 2: Divide the Leading Terms

Now divide the first term of the dividend \(x^{4}\) by the first term of the divisor \(x^2\). This gives us \(x^{2}\). Write \(x^{2}\) on top of the line.

Step 3: Multiply and Subtract

Next, we multiply the quotient \(x^{2}\) by each term in the divisor and subtract the resulting product from the dividend. This will give us a new polynomial of lower degree.

Step 4: Repeat the Process

The process is repeated with the new polynomial and the divisor. Continue this process until the degree of the remaining polynomial (i.e., the remainder) is less than the degree of the divisor.

Step 5: Write the Final Result

The result of the polynomial division is the quotient found, plus any remaining polynomial (the remainder) divided by the divisor. If the remainder is 0, then the result is just the quotient.

Key Concepts

Polynomial Division

Polynomial division is similar to long division of numbers, but it involves dividing variables with exponents. When performing polynomial division, the goal is to divide a polynomial (known as the dividend) by another polynomial (known as the divisor) to obtain a quotient and possibly a remainder. This technique is especially useful for simplifying complex algebraic expressions, solving polynomial equations, and analyzing functions.

The process is initiated by aligning the dividend and divisor in descending order, ensuring all powers of the variable are accounted for, even if their coefficients are zero. To be precise, any missing degrees must be filled with a term whose coefficient is zero. The division starts with the highest degree terms and proceeds until the remainder term has a lower degree than the divisor. One thing to keep in mind is that, like numerical division, if the divisor is a factor of the dividend, the remainder will be zero, indicating a perfect division.

Descending Order of Exponents

Arranging polynomials in descending order of exponents is a vital step in polynomial division. This means starting with the term with the highest exponent (also known as the leading term) and writing down to the term with the lowest exponent (the constant term), which has an exponent of zero. The arrangement is essential for maintaining clarity during the division process, as each term is handled systematically from the leading term down to the constant.

For example, if a polynomial is given as 1 + 3x² + x⁴, it must be rearranged to x⁴ + 3x² + 1 before the division can commence. Failure to follow this order might lead to confusion and mistakes in the division process, possibly resulting in an incorrect quotient and remainder. Hence, it is as crucial as knowing multiplication tables in numerical division.

Dividing Polynomials Step by Step

Dividing polynomials step by step may appear complicated initially, but it follows a systematic approach that, once understood, offers a clear route to the solution. To explain this, let's review the steps of the polynomial division process:

• Step 1: Write the dividend and the divisor in descending order of their exponents.
• Step 2: Divide the leading term of the dividend by the leading term of the divisor, placing the result above the dividend as the start of the quotient.
• Step 3: Multiply the divisor by the new term in the quotient and subtract this product from the dividend, creating a new, simplified polynomial.
• Step 4: Repeat steps 2 and 3 with the new polynomial until the remainder is either zero or of a lower degree than the divisor.
• Step 5: Document the quotient as the final result, or if there's a remainder, include it as a fractional part of the result.

Each step reduces the degree of the polynomial until the division process is complete. Careful arithmetic and attention to detail are essential throughout this process to ensure each subtraction and multiplication step is accurate. This method of division is a cornerstone of algebra and is pivotal for mastering more advanced mathematical concepts.