Overview
This lecture covers how to factor quadratic trinomials of the form x² + bx + c, including step-by-step examples, identifying prime trinomials, and applying factoring to solve basic word problems.
Listing Integer Pairs (Factors)
- List all pairs of integers whose product is a given number (the constant term c in the trinomial).
- Example: Factors of 16 are (1,16), (2,8), (4,4).
- Repeat the process for different values like 4, 6, 8, 10, 12, 18, 20, 30, etc.
Factoring Quadratic Trinomials (x² + bx + c)
- To factor x² + bx + c, find two integers whose product is c and whose sum is b.
- The factored form is (x + m)(x + n), where m + n = b and m × n = c.
- If c is positive, both integers are positive or both are negative.
- If c is negative, one integer is positive and one is negative; the larger integer has the sign of b.
Worked Examples
- x² + 10x + 16: Factors of 16 are (2,8); 2+8=10, so (x + 2)(x + 8).
- x² - 9x + 18: Factors of 18 are (-3,-6); -3 + (-6) = -9, so (x - 3)(x - 6).
- x² - 2x - 24: Factors of -24 are (4, -6); 4 + (-6) = -2, so (x + 4)(x - 6).
- x² + 3x - 10: Factors of -10 are (-2, 5); -2 + 5 = 3, so (x - 2)(x + 5).
- x² + 3x + 3: Factors are (1, 3); 1 + 3 ≠3, so it is a prime trinomial (cannot be factored with integers).
Factoring Non-Quadratic Polynomials (Word Problem)
- For 4x³ + 16x² - 48x: Factor out 4x to get 4x(x² + 4x - 12).
- Factor x² + 4x - 12: Factors are (6, -2); 6 + (-2) = 4, so (x + 6)(x - 2).
- Final factorization: 4x(x + 6)(x - 2). These are the box dimensions.
Key Terms & Definitions
- Quadratic trinomial — A polynomial in the form x² + bx + c.
- Prime trinomial — A trinomial that cannot be factored using integer coefficients.
- Factoring — Writing a polynomial as a product of simpler polynomials.
Action Items / Next Steps
- Practice factoring quadratic trinomials using the steps above.
- Identify and list integer pairs for various constants.
- Solve similar word problems involving polynomial factoring.