Solved on Dec 14, 2023

Divide the multivariate polynomial $-27 u^{2} z^{3}+12 u^{3} z^{6}-8 u z^{2}$ by the monomial $-4 u^{2} z^{3}$ and simplify the result.

STEP 1

Assumptions
1. We are given a polynomial $-27 u^{2} z^{3}+12 u^{3} z^{6}-8 u z^{2}$ to be divided by a monomial $-4 u^{2} z^{3}$ .
2. We will perform the division term by term.
3. The division of two terms with exponents is done by subtracting the exponents of like bases.

STEP 2

Divide the first term of the polynomial by the monomial.
$\frac{-27 u^{2} z^{3}}{-4 u^{2} z^{3}}$

STEP 3

Cancel out the common factors in the numerator and the denominator.
$\frac{-27}{-4} \cdot \frac{u^{2}}{u^{2}} \cdot \frac{z^{3}}{z^{3}}$

STEP 4

Since $u^{2}$ and $z^{3}$ are the same in both the numerator and the denominator, they cancel out, leaving:
$\frac{-27}{-4}$

STEP 5

Simplify the fraction by dividing -27 by -4.
$\frac{-27}{-4} = \frac{27}{4} = 6.75$

STEP 6

Now, move on to the second term of the polynomial and divide it by the monomial.
$\frac{12 u^{3} z^{6}}{-4 u^{2} z^{3}}$

STEP 7

Divide the coefficients and subtract the exponents of like bases.
$\frac{12}{-4} \cdot u^{3-2} \cdot z^{6-3}$

STEP 8

Simplify the expression.
$-3 u^{1} z^{3}$

STEP 9

Now, divide the third term of the polynomial by the monomial.
$\frac{-8 u z^{2}}{-4 u^{2} z^{3}}$

STEP 10

Divide the coefficients and subtract the exponents of like bases, noting that when the exponent in the denominator is larger, the result will be in the denominator of the fraction.
$\frac{-8}{-4} \cdot u^{1-2} \cdot z^{2-3}$

STEP 11

Simplify the expression.
$2 \cdot \frac{1}{u} \cdot \frac{1}{z}$

STEP 12

Combine the results from steps 5, 8, and 11 to write the final simplified expression.
$6.75 - 3u z^{3} + 2 \frac{1}{u z}$

STEP 13

Write the final answer as a simplified polynomial.
$\frac{27}{4} - 3u z^{3} + \frac{2}{u z}$
This is the simplified result of dividing the given polynomial by the given monomial.

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Divide the multivariate polynomial $-27 u^{2} z^{3}+12 u^{3} z^{6}-8 u z^{2}$ by the monomial $-4 u^{2} z^{3}$ and simplify the result.

STEP 1

Assumptions
1. We are given a polynomial $-27 u^{2} z^{3}+12 u^{3} z^{6}-8 u z^{2}$ to be divided by a monomial $-4 u^{2} z^{3}$ .
2. We will perform the division term by term.
3. The division of two terms with exponents is done by subtracting the exponents of like bases.

STEP 2

Divide the first term of the polynomial by the monomial.
$\frac{-27 u^{2} z^{3}}{-4 u^{2} z^{3}}$

STEP 3

Cancel out the common factors in the numerator and the denominator.
$\frac{-27}{-4} \cdot \frac{u^{2}}{u^{2}} \cdot \frac{z^{3}}{z^{3}}$

STEP 4

Since $u^{2}$ and $z^{3}$ are the same in both the numerator and the denominator, they cancel out, leaving:
$\frac{-27}{-4}$

STEP 5

Simplify the fraction by dividing -27 by -4.
$\frac{-27}{-4} = \frac{27}{4} = 6.75$

STEP 6

Now, move on to the second term of the polynomial and divide it by the monomial.
$\frac{12 u^{3} z^{6}}{-4 u^{2} z^{3}}$

STEP 7

Divide the coefficients and subtract the exponents of like bases.
$\frac{12}{-4} \cdot u^{3-2} \cdot z^{6-3}$

STEP 8

Simplify the expression.
$-3 u^{1} z^{3}$

STEP 9

Now, divide the third term of the polynomial by the monomial.
$\frac{-8 u z^{2}}{-4 u^{2} z^{3}}$

STEP 10

Divide the coefficients and subtract the exponents of like bases, noting that when the exponent in the denominator is larger, the result will be in the denominator of the fraction.
$\frac{-8}{-4} \cdot u^{1-2} \cdot z^{2-3}$

STEP 11

Simplify the expression.
$2 \cdot \frac{1}{u} \cdot \frac{1}{z}$

STEP 12

Combine the results from steps 5, 8, and 11 to write the final simplified expression.
$6.75 - 3u z^{3} + 2 \frac{1}{u z}$

STEP 13

Write the final answer as a simplified polynomial.
$\frac{27}{4} - 3u z^{3} + \frac{2}{u z}$
This is the simplified result of dividing the given polynomial by the given monomial.

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Divide the multivariate polynomial −27u2z3+12u3z6−8uz2-27 u^{2} z^{3}+12 u^{3} z^{6}-8 u z^{2}−27u2z3+12u3z6−8uz2 by the monomial −4u2z3-4 u^{2} z^{3}−4u2z3 and simplify the result.

STEP 1

STEP 2

Divide the first term of the polynomial by the monomial.−27u2z3−4u2z3\frac{-27 u^{2} z^{3}}{-4 u^{2} z^{3}}−4u2z3−27u2z3​

STEP 3

Cancel out the common factors in the numerator and the denominator.−27−4⋅u2u2⋅z3z3\frac{-27}{-4} \cdot \frac{u^{2}}{u^{2}} \cdot \frac{z^{3}}{z^{3}}−4−27​⋅u2u2​⋅z3z3​

STEP 4

Since u2u^{2}u2 and z3z^{3}z3 are the same in both the numerator and the denominator, they cancel out, leaving:−27−4\frac{-27}{-4}−4−27​

STEP 5

Simplify the fraction by dividing -27 by -4.−27−4=274=6.75\frac{-27}{-4} = \frac{27}{4} = 6.75−4−27​=427​=6.75

STEP 6

Now, move on to the second term of the polynomial and divide it by the monomial.12u3z6−4u2z3\frac{12 u^{3} z^{6}}{-4 u^{2} z^{3}}−4u2z312u3z6​

STEP 7

Divide the coefficients and subtract the exponents of like bases.12−4⋅u3−2⋅z6−3\frac{12}{-4} \cdot u^{3-2} \cdot z^{6-3}−412​⋅u3−2⋅z6−3

STEP 8

Simplify the expression.−3u1z3-3 u^{1} z^{3}−3u1z3

STEP 9

Now, divide the third term of the polynomial by the monomial.−8uz2−4u2z3\frac{-8 u z^{2}}{-4 u^{2} z^{3}}−4u2z3−8uz2​

STEP 10

Divide the coefficients and subtract the exponents of like bases, noting that when the exponent in the denominator is larger, the result will be in the denominator of the fraction.−8−4⋅u1−2⋅z2−3\frac{-8}{-4} \cdot u^{1-2} \cdot z^{2-3}−4−8​⋅u1−2⋅z2−3

STEP 11

Simplify the expression.2⋅1u⋅1z2 \cdot \frac{1}{u} \cdot \frac{1}{z}2⋅u1​⋅z1​

STEP 12

Combine the results from steps 5, 8, and 11 to write the final simplified expression.6.75−3uz3+21uz6.75 - 3u z^{3} + 2 \frac{1}{u z}6.75−3uz3+2uz1​

STEP 13