Factoring Quadratic Expressions: Basic and Advanced

Apuntes

• Factoring Quadratic Expressions: Basic and Advanced We start with the quadratic expression \( 2P^2 + 3P - 5 \). To factor it, we look for two binomials whose product gives the original quadratic. The process involves finding factors of the product of the quadratic coefficient (2) and the constant term (-5), which is -10. The factorization steps are as follows: \[ 2P^2 + 3P - 5 \\ = \frac{(2P + 5)(2P - 2)}{2} \\ = \frac{(2P + 5)^2 (P - 1)}{(2P + 5)(P - 1)} \\ = (2P + 5)(P - 1) \] The divisors of 12 are listed as: \[ D(12) = \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \] Similarly, the divisors of 10 are: \[ D(10) = \pm 1, \pm 2, \pm 5, \pm 10 \] --- • Factoring Trinomials: Basic Cases We continue with the process of factoring trinomials, starting with the simple form \( x^2 + bx + c \). **Example 1:** \[ x^2 + 5x + 6 \\ = (x + 2)(x + 3) \] **Example 2:** \[ t^3 - 3t^2 - 4t \] Factor out \( t \): \[ t(t^2 - 3t - 4) \] Factor the quadratic: \[ t(t - 4)(t + 1) \] **Example 3:** \[ K^2 + 5K + 6 \\ = (K + 2)(K + 3) \] --- • Factoring Trinomials with \( a \neq 1 \) **Example 2:** \[ t^3 - 3t^2 - 10t \] Factor out \( t \): \[ t(t^2 + 3t - 10) \] Now, factor the quadratic: \[ t(t + 5)(t - 2...