Factoring Trinomials

• Factoring trinomials involves finding two binomials that, when multiplied together, result in the original trinomial.
• The factored form of x² + 5x + 6 is (x + 2)(x + 3)
• The factored form of x² - 7x + 12 is (x - 4)(x - 3).
• The factored form of x² + 3x - 10 is (x + 5)(x - 2).
• The factored form of x² - x - 6 is (x - 3)(x + 2).
• The factored form of 2x² + 7x + 3 is (2x + 1)(x + 3).
• The factored form of x² - 9x + 20 is (x - 5)(x - 4).
• The factored form of x² + 8x + 16 is (x + 4)².
• The factored form of x² - 4x - 12 is (x - 6)(x + 2).
• The factored form of x² - 16x + 64 is (x - 8)².
• The factored form of 3x² + 11x + 10 is (3x + 5)(x + 2).
• Users can be tested on their ability to recognize how trinomials factor into binomials via a quiz.
• Such quizzes can test users by matching trinomials to their factored forms.
• Factoring trinomials involves expressing a trinomial as a product of two binomials.
• The factored form of x² + 7x + 12 is (x + 3)(x + 4).
• The factored form of x² - 5x + 6 is (x - 2)(x - 3).
• The factored form of 2x² + 11x + 12 is (2x + 3)(x + 4).
• The factored form of x² - x - 12 is (x + 3)(x - 4).

General Form of a Trinomial

• A trinomial is a polynomial with three terms.
• The general form of a trinomial is ax² + bx + c, where a, b, and c are constants, and x is a variable.

Factoring Trinomials when a = 1

• When the coefficient of x² is 1 (i.e., a = 1), the trinomial is in the form x² + bx + c.
• To factor x² + bx + c, find two numbers that multiply to c and add up to b.
• If these two numbers are p and q, then x² + bx + c = (x + p)(x + q).
• To factor x² + 5x + 6, find two numbers that multiply to 6 and add up to 5.
• The numbers are 2 and 3, since 2 * 3 = 6 and 2 + 3 = 5.
• Therefore, x² + 5x + 6 = (x + 2)(x + 3).

Factoring Trinomials when a ≠ 1

• When the coefficient of x² is not 1 (i.e., a ≠ 1), the trinomial is in the form ax² + bx + c.
• There are several methods to factor such trinomials, including trial and error, decomposition, or using the quadratic formula.

Factoring by Trial and Error

• Trial and error involves testing different combinations of factors until the correct factorization is found.
• This method is more efficient when a and c have fewer factors.
• Possible factor pairs for 2x² are (2x, x).
• Possible factor pairs for 3 are (3, 1).
• Test combinations: (2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3.
• For example, 2x² + 7x + 3 = (2x + 1)(x + 3).

Factoring by Decomposition (or AC Method)

• Multiply a and c.
• Find two numbers that multiply to ac and add up to b.
• Rewrite the middle term (bx) using these two numbers.
• Factor by grouping.
• To factor 2x² + 7x + 3, multiply a and c: 2 * 3 = 6.
• Find two numbers that multiply to 6 and add up to 7; the numbers are 6 and 1.
• Rewrite the middle term: 2x² + 6x + 1x + 3.
• Factor by grouping: 2x(x + 3) + 1(x + 3).
• Factor out the common binomial: (2x + 1)(x + 3).
• Thus, 2x² + 7x + 3 = (2x + 1)(x + 3).

Using the Quadratic Formula

• The quadratic formula can be used to find the roots of the quadratic equation ax² + bx + c = 0.
• The roots are given by x = (-b ± √(b² - 4ac)) / (2a).
• If the roots are x1 and x2, then the factored form of the trinomial is a(x - x1)(x - x2).
• Using the quadratic formula for x² - 5x + 6: x = (5 ± √((-5)² - 4 * 1 * 6)) / (2 * 1) = (5 ± √1) / 2.
• The roots are x1 = (5 + 1) / 2 = 3 and x2 = (5 - 1) / 2 = 2.
• Therefore, x² - 5x + 6 = (x - 3)(x - 2).

Recognizing Special Cases

• Perfect Square Trinomials: a² + 2ab + b² = (a + b)² and a² - 2ab + b² = (a - b)².
• Difference of Squares: a² - b² = (a + b)(a - b).

Factoring a Perfect Square Trinomial

• To factor x² + 6x + 9 recognize that x² is a square, 9 is a square (3²), and 6x is 2 * x * 3.
• Therefore, x² + 6x + 9 = (x + 3)².

Factoring a Difference of Squares

• To factor x² - 4, know that this is a difference of squares because x² and 4 are both squares.
• Therefore, x² - 4 = (x + 2)(x - 2).

Steps for Factoring Trinomials

• Look for a common factor in all terms and factor it out first.
• If the trinomial is in the form x² + bx + c (a = 1), find two numbers that multiply to c and add up to b.
• If the trinomial is in the form ax² + bx + c (a ≠ 1), use trial and error, decomposition, or the quadratic formula.
• Check if the trinomial is a perfect square trinomial or a difference of squares.
• Always check factorization by multiplying the binomials to ensure they result in the original trinomial.

Comprehensive Factoring Example

• To factor 3x² + 12x + 9, first factor out the common factor 3: 3(x² + 4x + 3).
• Now, factor the trinomial x² + 4x + 3 by finding two numbers that multiply to 3 and add up to 4.
• The numbers are 1 and 3.
• Therefore, x² + 4x + 3 = (x + 1)(x + 3).
• The complete factorization is 3(x + 1)(x + 3).

Tips for Factoring

• Practice regularly to improve factoring skills.
• Pay attention to signs, as they are crucial in determining the correct factors.
• Use the FOIL (First, Outer, Inner, Last) method to check factorization: (x + p)(x + q) = x² + qx + px + pq = x² + (p + q)x + pq.
• If a trinomial cannot be factored using integers, it is considered prime or irreducible over the integers.

Factoring and Solving Quadratic Equations

• Factoring is often used to solve quadratic equations of the form ax² + bx + c = 0.
• Factor the quadratic expression into two binomials: (x + p)(x + q) = 0.
• Set each factor equal to zero and solve for x: x + p = 0 or x + q = 0.
• The solutions are x = -p and x = -q, which are the roots or zeros of the quadratic equation.

Learn how to factor trinomials by finding two binomials. Examples of factoring trinomials, including expressions with positive and negative coefficients. Includes general strategies.