Please can someone help me? Thanks!
Find the least common multiple of each pair of polynomials:
12x^2-6x-126 and 18x-63
Simplify each sum:
5y+2 2x-4
---- + -----
xy^2 4xy
To find the least common multiple (LCM) of the polynomials 12x^2 - 6x - 126 and 18x - 63, you can follow these steps:
Step 1: Factorize each polynomial.
- The first polynomial, 12x^2 - 6x - 126, can be factored as 6(2x - 7)(x + 3).
- The second polynomial, 18x - 63, can be factored as 9(2x - 7).
Step 2: Determine the highest power of each factor.
- From the first polynomial, the highest power of 2x - 7 is 1, and the highest power of x + 3 is 1.
- From the second polynomial, the highest power of 2x - 7 is 1.
Step 3: Multiply the highest powers of each factor.
- The LCM will be 6 * (2x - 7) * (x + 3) * 9 = 54(2x - 7)(x + 3).
Therefore, the least common multiple of the polynomials 12x^2 - 6x - 126 and 18x - 63 is 54(2x - 7)(x + 3).
Now, let's simplify the sum: (5y + 2)/(xy^2) + (2x - 4)/(4xy).
To combine these fractions, we need a common denominator.
The common denominator will be 4xy * xy^2 = 4x^2y^3.
Now we can rewrite the fractions with the common denominator:
[(5y + 2)(y)]/(xy^2 * y) + [(2x - 4)(x)]/(4xy * x).
Simplifying the numerators, we get:
(5y^2 + 2y)/(xy^2) + (2x^2 - 4x)/(4xy).
Now, combine the fractions:
(5y^2 + 2y + 2x^2 - 4x)/(xy^2).
Therefore, the simplified sum is (5y^2 + 2y + 2x^2 - 4x)/(xy^2).