Factoring Polynomials (Lesson 3.06)

Key Concept: Factor Out a Common Binomial Factor

• The greatest common factor (GCF) can be a binomial instead of solely a monomial.

• The process for factoring out a binomial GCF mirrors the method for a monomial GCF.

Example 1

• Evaluate the expression:

  • Expression: $5xy - 16y$

  • GCF: $y$

  • Factored form: $y(5x - 16)$

Example 2

• Evaluate the expression:

  • Expression: $5x(x + 3) - 16(x + 3)$

  • GCF: $(x + 3)$

  • Factored form: $(x + 3)(5x - 16)$

Practice Problems

• Find the binomial GCF for each polynomial and factor completely:

  1. Polynomial: $4x(x - 1) + 9(x - 1)$

     • GCF: $(x - 1)$

     • Factored form: $(x - 1)(4x + 9)$

  2. Polynomial: $x(x + 7) - (x + 7)$

     • GCF: $(x + 7)$

     • Factored form: $(x + 7)(x - 1)$

  3. Polynomial: $x^2(x + 2) - 25(x + 2)$

     • GCF: $(x + 2)$

     • Factored form:

     • $(x + 2)(x^2 - 25)$

     • Further factoring:

     • $(x + 2)(x + 5)(x - 5)$

Factor by Grouping

• To factor a polynomial with four terms, use factor by grouping.

Example 3: Factor the polynomial $x^3 + 2x^2 - 25x - 50$ completely.

Step 1:

• Identify monomial GCF.

• No GCF present; retain original polynomial: $x^3 + 2x^2 - 25x - 50$

Step 2:

• Group terms into two pairs:

  • Grouping: $(x^3 + 2x^2) + (-25x - 50)$

Step 3:

• Find GCF for each pair:

  • First pair GCF: $x^2$

  • Second pair GCF is negative: $-25$

  • Factored form: $x^2(x + 2) - 25(x + 2)$

Step 4:

• Identify common binomial factor.

• Result: $(x + 2)$

Step 5:

• Factor out common binomial:

  • Resulting in: $(x + 2)(x^2 - 25)$

Step 6:

• Factor completely:

  • $(x + 2)(x - 5)(x + 5)$

Step 7:

• Check by multiplying the factors back together to verify correctness.

  • Original polynomial:

    • $x^3 + 2x^2 - 25x - 50$

Summary of Steps to Factor by Grouping

1. Always look for a GCF first. If there is one, factor it out.

2. Group the four terms into two pairs: (first and second terms, third and fourth terms).

3. Factor GCF out of each pair.

   • If a pair has no GCF, factor out a 1.

   • If the leading coefficient is negative, factor out a negative GCF.

4. If no common binomial arises, rearrange terms and restart.

5. Factor out the common binomial factor.

6. Factor completely. Factors with exponents may factor again.

7. Verify by multiplying factors back to the original polynomial.

Factoring Example

• Factor completely:

  • Expression: $x^2(x - 4) + 4(x - 4)$

  • Factored form is:

  • $(x - 4)(x^2 + 4)$

  • Reasoning: Binomial $(x - 4)$ is common; works like a monomial GCF.

  • Note: $(x^2 + 4)$ cannot be factored over real numbers; note the significance of factoring using complex numbers if required.

Further Factorization Example

Expression:

• $12ab + 12a9a^2 + 16$

Method:

1. Notice four terms: try factoring by grouping.

2. Factor out GCF:

   • $(12ab + 12a) + (9a^2 + 16)$

3. Rearrange original terms:

   • Result:

     • $(12ab + 9a^2) + (16 + 12a)$

4. GCF from pairs: $(12a(b + 1) + 1(9a^2 + 16))$

5. Binomials not matching, unable to continue; rearrange terms and regroup.

6. Resulting GCF is $12a$; second pair does not have a GCF.

7. Further experimentation with pairing: $3a(4b + 3a) + 4(4b + 3a)$

8. Factor out common binomial:

   • Result: $(4b + 3a)(3a + 4)$

9. Finalizing: Each polynomial is completely factored.

More Examples and Key Concepts for Trinomials

• Trinomial of the form $ax^2 + bx + c$ where leading coefficient $a ≠ 1$:

Steps:

1. Check for GCF.

2. Identify coefficient $b$.

3. Calculate product $ac$.

4. Find two factors of $ac$ whose sum is $b$.

5. Split the middle term into two terms.

6. Factor by grouping.

7. Ensure final factorization is complete.

Conclusion: The process outlined leads to an effective method for factoring polynomials through various strategies, including binomial GCFs, grouping, and recognizing patterns in trinomials. Check through multiplication helps confirm accuracy in the properties provided above.