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Goal: Factor trinomials of the form ax2 + bx + c
10.6 Factoring ax2 + bx + c Goal: Factor trinomials of the form ax2 + bx + c Standard: 11.0, 14.0, 15.0, 23.0

Standard 11.0 Students apply basic factoring techniques to second-and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials. Los estudiantes aplican técnicas básicas de factorización a polinómios de segundo y tercer grado. Estas técnicas incluyen encontrar un factor común para todos los términos del polinómio, reconociendo la diferencia de dos cuadrados, y reconociendo cuadrados perfectos de los binómios.

Standard 14.0 Students solve a quadratic equation by factoring or completing the square. Los estudiantes solucionan una equación cuadrática descomponiendola en factores o completando el cuadrado.

Standard 15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems. Los estudiantes aplican técnicas algebráicas para solucionar problemas de fracciones, problemas del trabajo, y problemas de porcentajes y mezcla de los anteriores.

Standard 23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity. Los estudiantes aplican equaciones cuadráticas a los problemas de física, tales como el movimiento de un objeto bajo fuerza de la gravedad.

Factor a trinomial Means to write it as the product of two binomials.
To Factor x2 + bx + c, you need to find numbers p and q such that P + q = b (linear coefficient) and, Pq = c (y-intercept) The difference between x2 + bx + c and ax2 + bx + c is that the leading coefficient is different than one (1), but the math process is still similar.

Example 1 Factor 3x2 + 7x + 2 A = 3 (leading coefficient, parabola opens upwards) B = 7 (linear coefficient) C = 2 (y-intercept when x = 0)

Solution Multiply each term of the quadratic trinomial by the same leading coefficient (3) 3x2 + (3) 7x + (3) 2 9x2 + (7) 3x + 6 switch the 2nd. term 7 goes outside and 3 inside Find two factors that give you your y-intercept or constant 6 6 = 1(6), 2(3) multiply Add the previous numbers to find the linear coefficient 7 7 = 1 + 6 x2 + 6x + 8 = (x + 4)(x + 2)

Solution 9x2 + (7) 3x + 6 Switch the 2nd. term 7 goes outside and 3 inside 6 = 1(6), 2(3) Multiply 7 = Add Square root the 1st. Monomial Open two binomials and each one write the result of the square root (3x )(3x ) The 1st. sign of the quadratic trinomial is positive (+) the that it is the sign of the 1st. binomial.

Solution (3x )(3x ) Multiply the signs of the trinomial: positive (+) times positive (+) is positive (+), which is the 2nd. Sign of the 2nd binomial (3x + 6)(3x + 1) Because at the beginning of the solution we multiply each term of the quadratic trinomial by the leading coefficient, then we have to undo this operation by dividing by the same leading coefficient 3 3x2 + 7x + 2 = (x + 2)(3x + 1)

Graphic Calculator Y-intercept 2 when x = 0 (x+2)=0, then x=-2
It is the 1st. solution Y-intercept 2 when x = 0 (3x+1)=0, then x=-1/3 It is the 2nd. solution

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