Example Question - algebraic expression factorization
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Factoring a Difference of Squares and Prime Factors
Para resolver este problema, primero debemos factorizar la diferencia de cuadrados que se presenta en la expresión \( P(x,y) = (5x + 3y)^2 - (x - 2y)^2 \). La factorización de una diferencia de cuadrados tiene la forma general \( a^2 - b^2 = (a + b)(a - b) \), donde \( a = 5x + 3y \) y \( b = x - 2y \). Utilizando esta identidad, factorizamos la expresión: \[ P(x,y) = (5x + 3y + x - 2y)(5x + 3y - x + 2y) \] \[ P(x,y) = ((5x + x) + (3y - 2y))((5x - x) + (3y + 2y)) \] \[ P(x,y) = (6x + y)(4x + 5y) \] Ahora, para encontrar la suma de los factores primos, primero identificamos si \( 6x + y \) y \( 4x + 5y \) pueden ser factorizados en términos de números primos. Sin embargo, como estas expresiones incluyen variables, no podemos factorizarlas en factores primos numéricos. Los factores primos se refieren generalmente a números y no a expresiones algebraicas. Por lo tanto, la suma de factores primos no tiene sentido en este contexto, dado que los factores que hemos obtenido dependen de las variables \( x \) y \( y \), y no son números específicos los cuales podríamos descomponer en primos.
Algebraic Expression Factorization
The problem in the image is asking to factorize the given algebraic expression. Let's do this step-by-step. First, we have the expression: (x + 1) / (x - 2) + (x - 1) / (x^2 - 4) The second denominator, x^2 - 4, is a difference of squares and can be factored as: x^2 - 4 = (x + 2)(x - 2) Now let's rewrite the expression with this factored form: (x + 1) / (x - 2) + (x - 1) / [(x + 2)(x - 2)] To combine these fractions, they need to have a common denominator. The least common denominator is (x + 2)(x - 2). So, we multiply the numerator and denominator of the first fraction by (x + 2) to have the same denominator for both terms: [(x + 1)(x + 2)] / [(x - 2)(x + 2)] + (x - 1) / [(x + 2)(x - 2)] Expanding the numerator of the first term, we get: (x^2 + 3x + 2) / [(x - 2)(x + 2)] + (x - 1) / [(x + 2)(x - 2)] Now that we have a common denominator, we can combine the numerators: (x^2 + 3x + 2 + x - 1) / [(x + 2)(x - 2)] Combining like terms in the numerator: (x^2 + 4x + 1) / [(x + 2)(x - 2)] This is the final factorized form of the given expression. No further simplification is possible unless we further factor the numerator, which in this case, does not factor nicely over the integers or rationals.
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