Factorizar agrupando (también llamado el método $ac$).
$3 x^2+11 x+10$
Primero, seleccionar una forma con los signos apropiados.
Luego, llenar los blancos con los números que va a utilizar para agrupar. Finalmente, mostrar la factorización.
Forma:
$3 x^2+\square x+\square x+10$
$3 x^2+\square x-\square x+10$
$3 x^2-\square x+\square x+10$
$3 x^2-\square x-\square x+10$
Factorización:
$\square$
Answer (2)
Find two numbers that multiply to 3 × 10 = 30 and add up to 11 . These numbers are 5 and 6 .
Rewrite the middle term: 3 x 2 + 5 x + 6 x + 10 .
Factor by grouping: ( 3 x 2 + 6 x ) + ( 5 x + 10 ) = 3 x ( x + 2 ) + 5 ( x + 2 ) .
Factor out the common binomial factor: ( 3 x + 5 ) ( x + 2 ) . The factorization is ( 3 x + 5 ) ( x + 2 ) .
Explanation
Understanding the Problem We are given the quadratic expression 3 x 2 + 11 x + 10 and asked to factor it using the grouping method (also known as the 'ac' method). Our goal is to rewrite the middle term, 11 x , as a sum of two terms such that we can then factor by grouping. First, we need to find two numbers that multiply to 3 × 10 = 30 and add up to 11 .
Finding the Right Numbers The two numbers that satisfy these conditions are 5 and 6 , since 5 × 6 = 30 and 5 + 6 = 11 . Now we can rewrite the middle term using these two numbers: 3 x 2 + 5 x + 6 x + 10 .
Factoring by Grouping Next, we factor by grouping. We group the first two terms and the last two terms: ( 3 x 2 + 6 x ) + ( 5 x + 10 ) . Now, we factor out the greatest common factor (GCF) from each group. From the first group, 3 x 2 + 6 x , the GCF is 3 x , so we have 3 x ( x + 2 ) . From the second group, 5 x + 10 , the GCF is 5 , so we have 5 ( x + 2 ) . Thus, we have 3 x ( x + 2 ) + 5 ( x + 2 ) .
Final Factorization Now we factor out the common binomial factor ( x + 2 ) from the entire expression: ( 3 x + 5 ) ( x + 2 ) . Therefore, the factorization of 3 x 2 + 11 x + 10 is ( 3 x + 5 ) ( x + 2 ) . The correct form is 3 x 2 + □ x + □ x + 10 , and the numbers are 5 and 6 .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For example, suppose you are designing a rectangular garden and you know the area can be represented by the expression 3 x 2 + 11 x + 10 , where x is a variable related to the dimensions. By factoring this expression into ( 3 x + 5 ) ( x + 2 ) , you determine the possible dimensions of the garden in terms of x . This allows you to plan the layout and optimize the use of space, making sure your garden fits perfectly in your yard.
Utilizando el método de agrupación, factoricé la expresión cuadrática 3 x 2 + 11 x + 10 en ( 3 x + 5 ) ( x + 2 ) . Los números que encontré son 5 y 6 . La factorización es ( 3 x + 5 ) ( x + 2 ) .
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