What is lcm 3z^3-6z^2-9z,7z^4+21z^3+14z^2 ?

The solution to lcm 3z^3-6z^2-9z,7z^4+21z^3+14z^2 is 21z^2(z+1)(z+2)(z-3)

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Popular Pre Algebra >

lcm 3z^3-6z^2-9z,7z^4+21z^3+14z^2

Solution

lcm

Solution

21 z^{2} (z + 1) (z + 2) (z - 3)

Solution steps

Find Least Common Multiplier of

3 z^{3} - 6 z^{2} - 9 z, 7 z^{4} + 21 z^{3} + 14 z^{2}

Factor

3 z^{3} - 6 z^{2} - 9 z : 3 z (z + 1) (z - 3)

3 z^{3} - 6 z^{2} - 9 z

Factor out common term

3 z : 3 z (z^{2} - 2 z - 3)

3 z^{3} - 6 z^{2} - 9 z

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

z^{2} = zz

= 3 z^{2} z - 6 zz - 9 z

Rewrite

9

3 \cdot 3

Rewrite

6

3 \cdot 2

= 3 z^{2} z - 3 \cdot 2 zz - 3 \cdot 3 z

Factor out common term

3 z

= 3 z (z^{2} - 2 z - 3)

= 3 z (z^{2} - 2 z - 3)

Factor

z^{2} - 2 z - 3 : (z + 1) (z - 3)

z^{2} - 2 z - 3

Break the expression into groups

z^{2} - 2 z - 3

Definition

Factors of

3 : 1, 3

3

Divisors (Factors)

Find the Prime factors of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Add 1

1

The factors of

3

1, 3

Negative factors of

3 : - 1, - 3

Multiply the factors by

- 1

to get the negative factors

- 1, - 3

For every two factors such that

u * v = - 3,

check if

u + v = - 2

Check

u = 1, v = - 3 : u * v = - 3, u + v = - 2 \Rightarrow

TrueCheck

u = 3, v = - 1 : u * v = - 3, u + v = 2 \Rightarrow

False

u = 1, v = - 3

Group into

(a x^{2} + ux) + (vx + c)

(z^{2} + z) + (- 3 z - 3)

= (z^{2} + z) + (- 3 z - 3)

Factor out

z

from

z^{2} + z : z (z + 1)

z^{2} + z

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

z^{2} = zz

= zz + z

Factor out common term

z

= z (z + 1)

Factor out

- 3

from

- 3 z - 3 : - 3 (z + 1)

- 3 z - 3

Factor out common term

- 3

= - 3 (z + 1)

= z (z + 1) - 3 (z + 1)

Factor out common term

z + 1

= (z + 1) (z - 3)

= 3 z (z + 1) (z - 3)

Factor

7 z^{4} + 21 z^{3} + 14 z^{2} : 7 z^{2} (z + 1) (z + 2)

7 z^{4} + 21 z^{3} + 14 z^{2}

Factor out common term

7 z^{2} : 7 z^{2} (z^{2} + 3 z + 2)

7 z^{4} + 21 z^{3} + 14 z^{2}

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

z^{3} = z z^{2}

= 7 z^{2} z^{2} + 21 z z^{2} + 14 z^{2}

Rewrite

14

7 \cdot 2

Rewrite

21

7 \cdot 3

= 7 z^{2} z^{2} + 7 \cdot 3 z z^{2} + 7 \cdot 2 z^{2}

Factor out common term

7 z^{2}

= 7 z^{2} (z^{2} + 3 z + 2)

= 7 z^{2} (z^{2} + 3 z + 2)

Factor

z^{2} + 3 z + 2 : (z + 1) (z + 2)

z^{2} + 3 z + 2

Break the expression into groups

z^{2} + 3 z + 2

Definition

Factors of

2 : 1, 2

2

Divisors (Factors)

Find the Prime factors of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Add 1

1

The factors of

2

1, 2

For every two factors such that

u * v = 2,

check if

u + v = 3

Check

u = 1, v = 2 : u * v = 2, u + v = 3 \Rightarrow

True

u = 1, v = 2

Group into

(a x^{2} + ux) + (vx + c)

(z^{2} + z) + (2 z + 2)

= (z^{2} + z) + (2 z + 2)

Factor out

z

from

z^{2} + z : z (z + 1)

z^{2} + z

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

z^{2} = zz

= zz + z

Factor out common term

z

= z (z + 1)

Factor out

2

from

2 z + 2 : 2 (z + 1)

2 z + 2

Factor out common term

2

= 2 (z + 1)

= z (z + 1) + 2 (z + 1)

Factor out common term

z + 1

= (z + 1) (z + 2)

= 7 z^{2} (z + 1) (z + 2)

Multiply each factor with the highest power:

3 \cdot 7 \cdot z^{2} \cdot (z + 1) \cdot (z + 2) \cdot (z - 3)

Simplify

21 z^{2} (z + 1) (z + 2) (z - 3)

Popular Examples

Frequently Asked Questions (FAQ)

What is lcm 3z^3-6z^2-9z,7z^4+21z^3+14z^2 ?
The solution to lcm 3z^3-6z^2-9z,7z^4+21z^3+14z^2 is 21z^2(z+1)(z+2)(z-3)

lcm 3z^3-6z^2-9z,7z^4+21z^3+14z^2

Solution

Solution

Popular Examples

Frequently Asked Questions (FAQ)

What is lcm 3z^3-6z^2-9z,7z^4+21z^3+14z^2 ?

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