1. find the least common multiple of x^2 +x -12 and x^2+2x -15
How ever did you arrive at that answer? Do try to be correct when you weigh in, and maybe show some of your work.
x^2+x-12 = (x+4)(x-3)
x^2+2x-15 = (x-3)(x+5)
so the LCM is (x+4)(x-3)(x+5) = x^3 + 6x^2 - 7x - 60
Step 1: Factor both polynomials.
The first polynomial x^2 + x - 12 can be factored as (x + 4)(x - 3).
The second polynomial x^2 + 2x - 15 can be factored as (x + 5)(x - 3).
Step 2: Identify the common factors.
Both polynomials share the common factor of (x - 3).
Step 3: Write the factored form of each polynomial without the common factors.
The first polynomial can be written as (x + 4).
The second polynomial can be written as (x + 5).
Step 4: Multiply the factored form of each polynomial.
Multiply (x + 4)(x + 5) to get (x^2 + 9x + 20).
Step 5: Multiply the common factor.
Multiply (x - 3) by (x^2 + 9x + 20) to get (x^3 + 6x^2 - 35x - 60).
Therefore, the least common multiple of x^2 + x - 12 and x^2 + 2x - 15 is x^3 + 6x^2 - 35x - 60.
To find the least common multiple (LCM) of the given polynomials, we first need to factorize them completely.
Both of the given polynomials are quadratic trinomials. We can factorize them by finding two binomials whose product is equal to the given quadratic trinomial.
1. Factorizing x^2 + x - 12:
We need to find two numbers whose product is -12 and whose sum is 1 (coefficient of x). After trying different combinations, we can factorize it as:
(x + 4)(x - 3)
2. Factorizing x^2 + 2x - 15:
We need to find two numbers whose product is -15 and whose sum is 2 (coefficient of x). After trying different combinations, we can factorize it as:
(x + 5)(x - 3)
Now that we have factorized both polynomials, we can find the LCM by taking the product of the highest powers of all the unique factors.
The unique factors in this case are: (x + 4), (x - 3), and (x + 5).
Taking the highest power of each factor, we get:
(x + 4)(x - 3)(x + 5)
Therefore, the least common multiple (LCM) of x^2 + x - 12 and x^2 + 2x - 15 is (x + 4)(x - 3)(x + 5).