Factoring Techniques and Examples

Factoring Techniques and Examples

Factoring with a Leading Coefficient (Guess and Check Method)

• Objective: Factor expressions like 2x^2 + 11x + 15.

• Method 1: Guess and check different combinations of factors.

  • Identify possible combinations to get the 2x^2 term (e.g., 2x and x).

  • List factor pairs of the constant term (e.g., 1 and 15, 3 and 5 for 15).

  • Try different arrangements of these factors and check if the middle term (11x in this case) is obtained when multiplying out.

  • Example: Trying (2x + 1)(x + 15) or (2x + 15)(x + 1), and checking if the outside and inside products add up to 11x.

Factoring with a Leading Coefficient (Alternative Method)

• Multiply the leading coefficient by the constant term (2 \times 15 = 30).

• Find factor pairs of the result (30) that add up to the middle coefficient (11).

  • In this case, 5 and 6 (5 \times 6 = 30 and 5 + 6 = 11).

• Rewrite the middle term using these factors (11x = 6x + 5x).

• Rewrite the expression: 2x^2 + 6x + 5x + 15.

• Factor by grouping:

  • 2x(x + 3) + 5(x + 3)

  • (2x + 5)(x + 3)

Example with Negative Numbers

• Example: 3x^2 - x - 4.

• Multiply the leading coefficient by the constant term: 3 \times -4 = -12.

• Find factor pairs of -12 that add up to -1.

  • 3 and -4 (3 - 4 = -1).

• Rewrite the middle term: 3x^2 + 3x - 4x - 4.

• Factor by grouping:

  • 3x(x + 1) - 4(x + 1)

  • (3x - 4)(x + 1)

• If the last sign in the expression is negative, the two factors will have different signs.

• If the last sign is positive, both factors will have the same sign (as the middle term).

Factoring Out Common Factors

• Always check if all terms have a common factor before applying other factoring techniques.

• Example: 12x^2y - 22xy + 8y.

• All terms are divisible by 2y. 2y(6x^2 - 11x + 4).

• If you want a specific sign inside the parenthesis, factor out the negative as well. Example: Factoring out -4. 4(-3x^2 + …).

Divisibility Rule for 3

• To check if a number is divisible by 3, add the digits of the number. If the sum is divisible by 3, then the original number is also divisible by 3.

• Example: 18 (1 + 8 = 9, which is divisible by 3).

Perfect Square Trinomials

• Recognizing perfect square trinomials can simplify factoring.

• Form: a^2 + 2ab + b^2 = (a + b)^2 or a^2 - 2ab + b^2 = (a - b)^2.

• Criteria:

  • The first term is a perfect square.

  • The last term is a perfect square, and the last sign is a plus.

  • The middle term is twice the product of the square roots of the first and last terms.

• Example: m^2 + 10m + 25.

  • m is the square root of m^2.

  • 5 is the square root of 25.

  • 2 \times m \times 5 = 10m (matches the middle term).

  • Factored form: (m + 5)^2.

• Example: b^2 + 16b + 64 = (b + 8)^2.

• Example: 12a^2x - 12abx + 3bx; after factoring out 3x, we get 3x(4a^2 - 4ab + b^2) = 3x(2a - b)^2

Difference of Two Squares

• Form: a^2 - b^2 = (a + b)(a - b).

• Criteria:

  • Two terms are perfect squares.

  • There is a minus sign between them.

• Example: x^2 - 9 = (x + 3)(x - 3).

• Example: 16y^2 - 9 = (4y + 3)(4y - 3).

• A sum of two squares (e.g., x^2 + y^2) does not factor using real numbers.

• Example: x^2 - \frac{1}{4} = (x + \frac{1}{2})(x - \frac{1}{2}).

Sum and Difference of Cubes

• Sum of Cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2).

• Difference of Cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2).

• SOPPS mnemonic:

  • S: Square the first term.

  • O: Opposite sign.

  • P: Product of the two terms.

  • P: Plus (always).

  • S: Square the second term.

• Example: x^3 + 8.

  • x is the cube root of x^3.

  • 2 is the cube root of 8.

  • Binomial: (x + 2).

  • Trinomial:

    • Square the first term: x^2.

    • Opposite sign: -

    • Product of the two terms: 2x.

    • Plus (always): +.

    • Square the second term: 4.

  • Factored form: (x + 2)(x^2 - 2x + 4).

• Example: n^3 - 125. After applying the formula: (n - 5)(n^2 + 5n + 25).

Solving Equations by Factoring

• If the product of two factors is zero, then at least one of the factors must be zero.

• Example: (x + 2)(x - 6) = 0. Set each factor equal to zero: x + 2 = 0 or x - 6 = 0. Solving each equation gives x = -2 or x = 6.

• To solve a quadratic equation, rearrange it so one side is zero, then factor the other side.

• Example: (2x^2 + 9x - 5) = 0. The factored form is (x+5)(2x-1) = 0, where x = -5 or x=1/2.

• If one side of an equation is not zero, rearrange it to make it zero before factoring.

  • Example: 2x(x - \frac{7}{2}) = 4. Then distribute the left side to get 2x^2 -7x = 4 and set the equation to zero, yielding the following 2x^2 -7x - 4 = 0.

  • Using the factoring method described earlier, (2x+1)(x-4) = 0, therefore, x = -1/2 or x = 4 .