Dividing Polynomials: Long Division and Synthetic Division

Dividing Polynomials: Long Division and Synthetic Division

Dividing a Polynomial by a Monomial

Dividing a polynomial by a monomial involves dividing each term of the polynomial individually by the monomial. This process uses exponent rules to simplify each term.

• Quotient Rule for Exponents: When dividing powers with the same base, subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$

• Zero Exponent Rule: Any nonzero number raised to the zero power equals one: $a^0 = 1$, where $a \neq 0$

• Example:

  • $\frac{x^8}{x^5} = x^{8-5} = x^3$

  • $\frac{6x^2 - 15x + 8}{3x} = \frac{6x^2}{3x} - \frac{15x}{3x} + \frac{8}{3x} = 2x - 5 + \frac{8}{3x}$

  • $\frac{10x^3 - 5x^2 + 20x}{5x} = 2x^2 - x + 4$

Dividing Polynomials: Long Division

When dividing a polynomial by another polynomial, use the long division method, similar to numerical long division. This process is systematic and allows for the division of higher-degree polynomials.

• Steps:

  1. Arrange both the dividend and divisor in descending powers of the variable.

  2. Divide the leading term of the dividend by the leading term of the divisor.

  3. Multiply the entire divisor by the result and subtract from the dividend.

  4. Repeat the process with the new polynomial until the degree of the remainder is less than the divisor.

• Example:

  • $\frac{6x^2 - 19x + 12}{3x - 5}$

  • $\frac{6x^2 + 11x - 2}{x + 2}$

  • $\frac{27x^3 + 8}{3x + 2}$

  • $\frac{5x^3 + 9x^2 - 10x + 2}{x + 3}$

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a binomial of the form x - c or x + c. It is efficient and reduces the complexity of the division process.

• Applicable Form: The divisor must be linear (degree 1), such as $x - c$ or $x + c$.

• Procedure:

  1. Write the coefficients of the dividend polynomial.

  2. Use the value of c (for $x - c$) or -c (for $x + c$).

  3. Carry down the first coefficient, multiply by c, add to the next coefficient, and repeat.

  4. The final row gives the coefficients of the quotient and the remainder.

• Example:

  • Divide $2x^3 - x^2 - 13x + 1$ by $x - 3$ using synthetic division.

  • Divide $4x^3 - 3x^2 + 6x + 5$ by $x - 1$.

  • Divide $x^4 - 2x^3 - 11x^2 + 34$ by $x + 2$.

  • Divide $y^3 - 8$ by $y - 2$.

Practice Exercises

Apply the division techniques to the following exercises:

• $\frac{4x^7y^4 + 8xy^2 + 4xy^3}{4xy^3}$

• $\frac{10x^3 - 5x^2 - 12x + 1}{2x - 1}$

• $\frac{20x^3 - 8x^2 + 5x + 5}{5x - 2}$

• $\frac{2x^3 - 6x^2 - 4}{x - 4}$

Exponent Rules Reference Table

The following table summarizes key exponent rules used in polynomial division:

Rule Name	Formula	Description
Quotient Rule	$\frac{a^m}{a^n} = a^{m-n}$	Subtract exponents when dividing same base
Zero Exponent	$a^0 = 1$	Any nonzero base to zero power is 1