We want to find the least common multiple (LCM) of the given pair of polynomials.

x^{2} - 9 x + 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.

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Substitute values

x = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 ( 1 ) ( 2 0 )}{2 ( 1 )}

▼

Simplify right-hand side

$- (- a) = a$

x = \frac{9 \pm ( - 9 ) ^{2} - 4 ( 1 ) ( 2 0 )}{2 ( 1 )}

Calculate power

x = \frac{9 \pm 8 1 - 4 ( 1 ) ( 2 0 )}{2 ( 1 )}

Identity Property of Multiplication

x = \frac{9 \pm 8 1 - 4 ( 2 0 )}{2}

x = \frac{9 \pm 8 1 - 8 0}{2}

Subtract term

x = \frac{9 \pm 1}{2}

Calculate root

x = \frac{9 \pm 1}{2}

The solutions for this equation are

x = \frac{9 \pm 1}{2} .

Let's separate them into the positive and negative cases.

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.

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Substitute values

x = \frac{- 1 \pm 1 ^{2} - 4 ( 1 ) ( - 3 0 )}{2 ( 1 )}

▼

Simplify right-hand side

$1^{a} = 1$

x = \frac{- 1 \pm 1 - 4 ( 1 ) ( - 3 0 )}{2 ( 1 )}

Identity Property of Multiplication

x = \frac{- 1 \pm 1 - 4 ( - 3 0 )}{2}

$- a (- b) = a \cdot b$

x = \frac{- 1 \pm 1 + 1 2 0}{2}

Add terms

x = \frac{- 1 \pm 1 2 1}{2}

Calculate root

x = \frac{- 1 \pm 1 1}{2}

The solutions for this equation are

x = \frac{- 1 \pm 1 1}{2} .

Let's separate them into the positive and negative cases.

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)) \Leftrightarrow (x - 5) (x + 6)

We can now write both expressions as the product of their prime factors.

x^{2} - 9 x + 20 x^{2} + x - 30 = (x - 5) (x - 4) = (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)

Exercise