We want to find the least common multiple (LCM) of the given pair of polynomials.
x^2-9x+20 and x^2+x-30
To do so, we first need to find the prime factors of each expression.
Since the degree of the first polynomial is 2, we can use the Quadratic Formula to help us factor.
The solutions for this equation are x= 9 ± 12. Let's separate them into the positive and negative cases.
x=9 ± 1/2
x_1=9+1/2
x_2=9-1/2
x_1=10/2
x_2=8/2
x_1=5
x_2=4
Knowing that x= 5 and x= 4 are the zeros of the trinomial, we can write the factored form of the quadratic expression.
(x- 5)(x- 4)
Let's factor the second expression. Since the degree of the second polynomial is also 2, we can use the Quadratic Formula.
The solutions for this equation are x= - 1 ± 112. Let's separate them into the positive and negative cases.
x=- 1 ± 11/2
x_1=- 1+11/2
x_2=- 1-11/2
x_1=10/2
x_2=- 12/2
x_1=5
x_2=- 6
Knowing that x= 5 and x= - 6 are the zeros of the trinomial, we can write the factored form of the quadratic expression.
(x- 5)(x-( - 6)) ⇔ (x-5)(x+6)
We can now write both expressions as the product of their prime factors.
x^2-9x+20 &= (x-5) (x-4)
x^2+x-30 &= (x-5) (x+6)
Finally, we write the product of the prime factors, each raised to the greatest power that occurs in their expression.
(x-5) (x-4) (x+6)
