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How do you factor a fractional polynomial?

Published in Factoring Polynomials 3 mins read

Factoring a fractional polynomial (an expression with fractional or negative exponents) primarily involves identifying and extracting the Greatest Common Factor (GCF), particularly the variable raised to the lowest power present in any term. This approach leverages the same factoring principles applied to polynomials with integer exponents.

Steps for Factoring Fractional Polynomials:

  1. Identify the Terms: Clearly define each term within the fractional polynomial.

  2. Find the GCF:

    • Variable Component: Determine the variable(s) common to all terms. Identify the smallest exponent of each common variable. This smallest exponent becomes the exponent of the GCF's variable component.
    • Coefficient Component: Find the greatest common factor of the coefficients of all terms.

  3. Extract the GCF: Factor out the GCF from each term in the polynomial. This involves dividing each term by the GCF. Remember that when dividing terms with exponents, you subtract the exponents.

  4. Write the Factored Expression: Express the polynomial as the GCF multiplied by the expression remaining after factoring.

Example:

Let's factor the expression: x^(1/2) + x^(3/2)

  1. Terms: The terms are x^(1/2) and x^(3/2).

  2. GCF:

    • Variable: Both terms contain x. The lowest exponent is 1/2. So, the GCF is x^(1/2).

  3. Extract:

    • x^(1/2) / x^(1/2) = 1
    • x^(3/2) / x^(1/2) = x^((3/2)-(1/2)) = x^(2/2) = x

  4. Factored Expression: x^(1/2) (1 + x)

Therefore, the factored form of x^(1/2) + x^(3/2) is x^(1/2) (1 + x).

Another Example (with Negative Exponents):

Factor: 2x^(-1/3) + 4x^(2/3)

  1. Terms: 2x^(-1/3) and 4x^(2/3)

  2. GCF:

    • Coefficients: GCF of 2 and 4 is 2.
    • Variable: Both terms contain x. The smallest exponent is -1/3. So, the GCF is 2x^(-1/3).

  3. Extract:

    • 2x^(-1/3) / 2x^(-1/3) = 1
    • 4x^(2/3) / 2x^(-1/3) = 2x^((2/3)-(-1/3)) = 2x^(3/3) = 2x

  4. Factored Expression: 2x^(-1/3) (1 + 2x)

Key Considerations:

  • Negative Exponents: A negative exponent indicates a reciprocal. For example, x^(-1) is equivalent to 1/x. Be mindful of this when determining the smallest exponent.
  • Fractional Exponents as Radicals: Remember that a fractional exponent represents a root. For example, x^(1/2) is the square root of x, and x^(1/3) is the cube root of x. Factoring can sometimes help simplify radical expressions.

By carefully identifying and extracting the GCF, you can effectively factor fractional polynomials, simplifying expressions and solving equations.

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