Dividing Polynomials Math Lib Answer Key

Dividing Polynomials: A Comprehensive Guide with Worked Examples

Dividing polynomials is a fundamental concept in algebra with wide-ranging applications in various fields, including calculus, engineering, and computer science. Mastering polynomial division is crucial for simplifying expressions, solving equations, and understanding more advanced mathematical concepts. This comprehensive guide will walk you through the process of polynomial division, covering long division, synthetic division, and providing numerous worked examples to solidify your understanding. We'll also explore the concept of remainders and their significance.

Understanding Polynomials

Before diving into division, let's refresh our understanding of polynomials. A polynomial is an expression consisting of variables (usually denoted by x), coefficients, and exponents, combined using addition, subtraction, and multiplication. The exponents must be non-negative integers.

Examples of Polynomials:

• 3x² + 2x - 5
• 5x⁴ - 7x² + 2
• x³ + 4x
• 7 (a constant is also a polynomial)

Key Terminology:

• Degree: The highest exponent in the polynomial. The degree of 3x² + 2x - 5 is 2.
• Coefficient: The numerical factor of a term. In 3x², the coefficient is 3.
• Term: A single expression in the polynomial (e.g., 3x², 2x, -5).
• Constant Term: The term without a variable (e.g., -5 in 3x² + 2x - 5).

Method 1: Polynomial Long Division

Polynomial long division is a method used to divide a polynomial by another polynomial of a lower or equal degree. It's analogous to long division with numbers.

Steps for Polynomial Long Division:

1. Arrange: Arrange both the dividend (the polynomial being divided) and the divisor (the polynomial doing the dividing) in descending order of their exponents. Fill in any missing terms with a coefficient of 0 (e.g., x³ + 2x + 1 becomes x³ + 0x² + 2x + 1).

2. Divide: Divide the leading term of the dividend by the leading term of the divisor. This becomes the first term of the quotient.

3. Multiply: Multiply the divisor by the first term of the quotient.

4. Subtract: Subtract the result from the dividend.

5. Bring Down: Bring down the next term from the dividend.

6. Repeat: Repeat steps 2-5 until you reach a remainder with a degree lower than the divisor.

Worked Example:

Divide (6x³ + 17x² + 27x + 20) by (3x + 4)

                2x² + 3x + 5
        -----------------------
3x + 4 | 6x³ + 17x² + 27x + 20
        - (6x³ + 8x²)
        -----------------------
                 9x² + 27x
             - (9x² + 12x)
        -----------------------
                         15x + 20
                     - (15x + 20)
        -----------------------
                               0

Therefore, (6x³ + 17x² + 27x + 20) ÷ (3x + 4) = 2x² + 3x + 5

Method 2: Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear divisor (a divisor of the form x - c, where c is a constant). It's significantly faster than long division but only works for linear divisors.

Steps for Synthetic Division:

1. Set up: Write the coefficients of the dividend in a row. To the left, write the value of 'c' (from x - c) but change its sign (if c = 2, use -2).

2. Bring Down: Bring down the first coefficient.

3. Multiply and Add: Multiply the brought-down coefficient by 'c' and add it to the next coefficient. Repeat this process for all coefficients.

4. Interpret: The last number is the remainder. The other numbers are the coefficients of the quotient, starting with one degree less than the dividend.

Worked Example:

Divide (2x³ - 5x² + 3x - 6) by (x - 2) (Here, c = 2)

2 | 2  -5   3  -6
  |    4  -2   2
  ----------------
    2  -1   1  -4

The quotient is 2x² - x + 1, and the remainder is -4. Therefore, (2x³ - 5x² + 3x - 6) ÷ (x - 2) = 2x² - x + 1 - 4/(x-2)

Remainder Theorem

The remainder theorem states that when a polynomial f(x) is divided by (x - c), the remainder is f(c). This provides a quick way to find the remainder without performing the full division.

Worked Example:

Find the remainder when (x³ + 2x² - 5x + 7) is divided by (x - 3).

Using the remainder theorem, we substitute x = 3 into the polynomial:

f(3) = (3)³ + 2(3)² - 5(3) + 7 = 27 + 18 - 15 + 7 = 37

The remainder is 37.

Factor Theorem

A direct consequence of the remainder theorem is the factor theorem. It states that (x - c) is a factor of a polynomial f(x) if and only if f(c) = 0. In other words, if the remainder is 0, then (x - c) is a factor.

Worked Example:

Is (x - 1) a factor of x³ - 3x² + 2x - 6?

We evaluate f(1):

f(1) = (1)³ - 3(1)² + 2(1) - 6 = 1 - 3 + 2 - 6 = -6

Since f(1) ≠ 0, (x - 1) is not a factor.

Dividing Polynomials with Higher-Degree Divisors

While synthetic division is limited to linear divisors, long division can handle divisors of any degree. The process remains the same; you just need to be more careful with the multiplication and subtraction steps.

Worked Example:

Divide (x⁴ - 3x³ + 2x² - 1) by (x² - x + 1)

               x² - 2x - 1
       -------------------------
x² - x + 1 | x⁴ - 3x³ + 2x² + 0x - 1
           - (x⁴ - x³ + x²)
       -------------------------
               -2x³ + x² + 0x
           - (-2x³ + 2x² - 2x)
       -------------------------
                    -x² + 2x - 1
                - (-x² + x - 1)
       -------------------------
                           x + 0

Therefore, the quotient is x² - 2x -1 and the remainder is x.

Applications of Polynomial Division

Polynomial division has numerous applications in various fields:

• Simplifying Rational Expressions: Dividing polynomials helps in simplifying complex rational expressions.
• Finding Roots of Polynomials: Polynomial division is often used in conjunction with the factor theorem to find the roots (or zeros) of polynomials.
• Partial Fraction Decomposition: In calculus, polynomial division is crucial in partial fraction decomposition, a technique used to integrate rational functions.
• Curve Fitting and Modeling: Polynomial division plays a role in fitting curves to data and creating mathematical models.
• Engineering and Computer Science: Polynomial division finds applications in areas like control systems, signal processing, and computer-aided design.

Conclusion

Mastering polynomial division, whether through long division or synthetic division, is a critical skill for success in algebra and related fields. By understanding the underlying principles and practicing with diverse examples, you can build a strong foundation for more advanced mathematical concepts. Remember to always check your work carefully, particularly when dealing with negative signs and complex expressions. The more you practice, the more proficient you'll become. This thorough understanding will unlock a deeper appreciation for the elegance and power of polynomial algebra.