Mastering Polynomial Factoring

3. Factoring Trinomials (ax^2 + bx + c)

Trinomials are polynomials with three terms. The method for factoring them depends on the leading coefficient, a.

Case 1: Leading Coefficient is 1 (a = 1)

When the trinomial is in the form x^2 + bx + c, you need to find two numbers that multiply to c and add up to b.

Example:

Factor the trinomial x^2 + 5x + 6.

• We need two numbers that multiply to 6 and add to 5.
• The numbers are 2 and 3 (since 2 × 3 = 6 and 2 + 3 = 5).

Result: (x + 2)(x + 3)

Case 2: Leading Coefficient is Not 1 (a ≠ 1)

When the trinomial is in the form ax^2 + bx + c, a common method is the "AC method".

1. Multiply a and c: Find the product of a and c.

2. Find two numbers: Find two numbers that multiply to the product ac and add up to b.

3. Rewrite the middle term: Rewrite the middle term, bx, using the two numbers you found.

4. Factor by grouping: Use the factoring by grouping method described earlier.

Example:

Factor the trinomial 2x^2 + 7x + 3.

1. Multiply a and c: 2 × 3 = 6.

2. Find two numbers that multiply to 6 and add to 7. The numbers are 1 and 6.

3. Rewrite the middle term: 2x^2 + 1x + 6x + 3.

4. Factor by grouping: (2x^2 + x) + (6x + 3) = x(2x + 1) + 3(2x + 1).

Result: (2x + 1)(x + 3)

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4. Special Factoring Formulas

Recognizing special patterns can be a shortcut to factoring.

Difference of Squares

This applies to binomials of the form a^2 - b^2.

Formula: a^2 - b^2 = (a - b)(a + b)

Example:

Factor x^2 - 9.

• This is a difference of squares where a = x and b = 3.

Result: (x - 3)(x + 3)

Sum of Cubes

This applies to binomials of the form a^3 + b^3.

Formula: a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Example:

Factor x^3 + 8.

• This is a sum of cubes where a = x and b = 2.

Result: (x + 2)(x^2 - 2x + 4)

Difference of Cubes

This applies to binomials of the form a^3 - b^3.

Formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Example:

Factor y^3 - 27.

• This is a difference of cubes where a = y and b = 3.

Result: (y - 3)(y^2 + 3y + 9)