Factoring Polynomials by Grouping

Factoring Polynomials

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials with four or more terms. The process involves grouping terms in pairs and factoring out the greatest common factor (GCF) from each group, then factoring further if possible.

• Definition: Factoring by grouping is a technique for rewriting a polynomial as a product of simpler polynomials by grouping terms and factoring out common factors.

• Key Steps:

  1. Group the terms in pairs.

  2. Factor out the GCF from each group.

  3. If the resulting binomials are the same, factor them out.

• Formula: For a polynomial of the form $ax + ay + bx + by$, grouping gives: $ax + ay + bx + by = (ax + ay) + (bx + by) = a(x + y) + b(x + y) = (a + b)(x + y)$

Example: Factoring $2x^2 - 3x + 8x - 12$ by Grouping

• Step 1: Group the terms: $(2x^2 - 3x) + (8x - 12)$

• Step 2: Factor out the GCF from each group:

  • $2x^2 - 3x = x(2x - 3)$

  • $8x - 12 = 4(2x - 3)$

• Step 3: Factor out the common binomial $(2x - 3)$: $x(2x - 3) + 4(2x - 3) = (x + 4)(2x - 3)$

• Final Answer: $2x^2 - 3x + 8x - 12 = (x + 4)(2x - 3)$

Prime Polynomials

If a polynomial cannot be factored over the integers, it is called prime.

• Definition: A prime polynomial is a polynomial that cannot be factored into polynomials of lower degree with integer coefficients.

• Example: $x^2 + 1$ is prime over the real numbers.

Summary Table: Factoring by Grouping

Step	Description	Example
1	Group terms in pairs	$(2x^2 - 3x) + (8x - 12)$
2	Factor out GCF from each group	$x(2x - 3) + 4(2x - 3)$
3	Factor out common binomial	$(x + 4)(2x - 3)$
4	If no common binomial, polynomial may be prime	e.g., $x^2 + 1$

Additional info: The original file appears to be a homework or test question asking students to factor a polynomial by grouping and to determine if the polynomial is prime if it cannot be factored.