Question

What is the least common multiple of 6x^2+39x-21 and 6x^2+54x+84?

Answer 1

`12x^3+102x^2+114x-84`

Explanation:

Given polynomials:

`\begin{cases} 6x^2+39x-21\\6x^2+54x+84 \end{cases}`

Factor the polynomials:

Polynomial 1

`\implies 6x^2+39x-21`

`\implies 3(2x^2+13x-7)`

`\implies 3(2x^2+14x-x-7)`

`\implies 3[2x(x+7)-1(x+7)]`

`\implies 3(2x-1)(x+7)`

Polynomial 2

`\implies 6x^2+54x+84`

`\implies 6(x^2+9x+14)`

`\implies 6(x^2+7x+2x+14)`

`\implies 6[x(x+7)+2(x+7)]`

`\implies 6(x+2)(x+7)`

`\implies 2 \cdot 3(x+2)(x+7)`

The lowest common multiplier (LCM) of two polynomials a and b is the smallest multiplier that is divisible by both a and b.

Therefore, the LCM of the two polynomials is:

`\implies 2 \cdot 3(x+7)(x+2)(2x-1)`

`\implies (6x^2+54x+84)(2x-1)`

`\implies 12x^3+108x^2+168x-6x^2-54x-84`

`\implies 12x^3+102x^2+114x-84`

Answer 2

12x² + 102x² + 114x - 84

Answer:

Solution Given:

1st term: 6x²+39x-21

Taking common

3(2x²+13x-7)

doing middle term factorization

3(2x²+14x-x-7)

3(2x(x+7)-1(x+7))

3(x+7)(2x-1)

2nd term: 6x²+54x+84

taking common

6(x²+9x+14)

doing middle term factorization

6(x²+7x+2x+14)

6(x(x+7)+2(x+7))

2*3(x+7)(x+2)

Now

Least common multiple = 2*3(x+7)(2x-1)(x+2)

2(x+2)(6x²+39x-21)

(2x+4)(6x²+39x-21)

2x(6x²+39x-21)+4(6x² + 39x-21)

12x³+78x² - 42x+4(6x² + 39x-21)

12x³+78x² - 42x + 24x² + 156x-84

12x³ + 102x²-42x + 156x - 84

12x² + 102x² + 114x - 84