6.2 Dividing Polynomials and the Remainder Theorem

Dividing polynomials and the Remainder Theorem are key tools for working with polynomial functions. These methods help us break down complex polynomials, find their roots, and understand their behavior. They're essential for solving polynomial equations and factoring higher-degree polynomials.

By mastering these techniques, we can tackle more advanced problems in polynomial theory. We'll use them to find factors, evaluate polynomials efficiently, and determine divisibility. These skills build a strong foundation for understanding polynomial functions and their properties.

Polynomial Long Division

Dividing Polynomials

Top images from around the web for Dividing Polynomials

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Dividing Polynomials View original

  Is this image relevant?

• 

  Dividing Polynomials | College Algebra View original

  Is this image relevant?

• 

  Polynomial Long Division | College Algebra View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Dividing Polynomials View original

  Is this image relevant?

• 

  Dividing Polynomials | College Algebra View original

  Is this image relevant?

1 of 3

Top images from around the web for Dividing Polynomials

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Dividing Polynomials View original

  Is this image relevant?

• 

  Dividing Polynomials | College Algebra View original

  Is this image relevant?

• 

  Polynomial Long Division | College Algebra View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Dividing Polynomials View original

  Is this image relevant?

• 

  Dividing Polynomials | College Algebra View original

  Is this image relevant?

1 of 3

• Long division of polynomials is a method used to divide a polynomial by another polynomial, resulting in a quotient and a remainder
• The process of long division of polynomials involves:
  1. Dividing the highest degree term of the dividend by the highest degree term of the divisor
  2. Multiplying the result by the divisor
  3. Subtracting the product from the dividend
  4. Repeating the process until the degree of the remainder is less than the degree of the divisor
• The final result of long division of polynomials is expressed as: $Dividend = (Divisor × Quotient) + Remainder$
• The degree of the quotient is equal to the difference between the degrees of the dividend and the divisor ($degree(Quotient) = degree(Dividend) - degree(Divisor)$)

Divisibility and Factors

• If the remainder is 0, the divisor is a factor of the dividend
  • For example, if $P(x) = (x - 2)(x^2 + 3x + 4)$, then $(x - 2)$ is a factor of $P(x)$
• Polynomial long division can be used to factor higher-degree polynomials by dividing the polynomial by a known factor and obtaining the quotient and remainder
  • If the remainder is 0, the divisor is a factor of the polynomial, and the quotient is the other factor

Remainder Theorem Applications

Evaluating Polynomials

• The Remainder Theorem states that the remainder of a polynomial $P(x)$ divided by $(x - a)$ is equal to $P(a)$, where $a$ is a real number
• To evaluate a polynomial function $P(x)$ at $x = a$ using the Remainder Theorem, substitute $a$ for $x$ in the polynomial
  • For example, if $P(x) = x^3 - 2x^2 + 3x - 4$ and $a = 2$, then $P(2) = 2^3 - 2(2^2) + 3(2) - 4 = 8 - 8 + 6 - 4 = 2$

Determining Divisibility

• If $P(a) = 0$, then $(x - a)$ is a factor of $P(x)$, and the polynomial is divisible by $(x - a)$
• The Remainder Theorem can be used to determine the divisibility of a polynomial by a linear factor without performing long division
  • For example, if $P(x) = x^3 - 2x^2 + 3x - 4$ and $a = 1$, then $P(1) = 1^3 - 2(1^2) + 3(1) - 4 = 1 - 2 + 3 - 4 = -2 ≠ 0$, so $(x - 1)$ is not a factor of $P(x)$
• The Remainder Theorem is a special case of the Polynomial Division Algorithm

Factoring Polynomials

Using Polynomial Division

• To solve polynomial equations using polynomial division, set the polynomial equal to 0 and divide by a known factor
  • If the remainder is 0, the divisor is a solution to the equation
• Polynomial division can be used in conjunction with other factoring techniques, such as the Rational Root Theorem and synthetic division, to factor and solve higher-degree polynomials

Factor Theorem

• The Factor Theorem states that $(x - a)$ is a factor of a polynomial $P(x)$ if and only if $P(a) = 0$
• The number of linear factors of a polynomial is equal to the degree of the polynomial, and the polynomial can be written as a product of its linear factors
  • For example, if $P(x) = x^3 - 6x^2 + 11x - 6$, then $P(1) = 0$, $P(2) = 0$, and $P(3) = 0$, so $(x - 1)$, $(x - 2)$, and $(x - 3)$ are factors of $P(x)$, and $P(x) = (x - 1)(x - 2)(x - 3)$

Remainders and Factors

Relationship between Remainders and Factors

• If a polynomial $P(x)$ is divided by $(x - a)$ and the remainder is 0, then $(x - a)$ is a factor of $P(x)$, and $a$ is a root or solution of the polynomial equation $P(x) = 0$
• The relationship between the remainder and factors of a polynomial is the foundation for the Rational Root Theorem, which helps determine the potential rational roots of a polynomial equation

Rational Root Theorem

• The Rational Root Theorem states that if a polynomial equation $a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 = 0$ has integer coefficients, then any rational solution $x = \frac{p}{q}$ (where $p$ and $q$ are integers with no common factors) must satisfy:
  1. $p$ is a factor of the constant term $a_0$
  2. $q$ is a factor of the leading coefficient $a_n$
• The Rational Root Theorem can be used to list the potential rational roots of a polynomial equation, which can then be tested using the Remainder Theorem or synthetic division