Math / Algebra

QuestionDivide and simplify. $\frac{3 n^{9}+27 n^{5}-9 n^{2}}{-3 n^{4}}$ $\frac{3 n^{9}+27 n^{5}-9 n^{2}}{-3 n^{4}}=$ $\square$ (Simplify your answer.)

Studdy Solution

STEP 1

1. The expression is a rational expression that needs to be simplified.
2. The numerator and the denominator can be simplified by factoring out common terms.
3. The division of polynomials involves simplifying each term separately.

STEP 2

1. Factor out the greatest common factor (GCF) from the numerator.
2. Simplify the division by dividing each term by the denominator.
3. Simplify the resulting expression.

STEP 3

Factor out the greatest common factor from the numerator $3n^9 + 27n^5 - 9n^2$ .
The GCF of the terms in the numerator is $3n^2$ .
$3n^9 + 27n^5 - 9n^2 = 3n^2(n^7 + 9n^3 - 3)$

STEP 4

Divide each term in the factored numerator by the denominator $-3n^4$ .
$\frac{3n^2(n^7 + 9n^3 - 3)}{-3n^4} = \frac{3n^2}{-3n^4} \times (n^7 + 9n^3 - 3)$

STEP 5

Simplify the fraction $\frac{3n^2}{-3n^4}$ .
$\frac{3n^2}{-3n^4} = -\frac{1}{n^2}$

STEP 6

Multiply the simplified fraction by the remaining polynomial.
$-\frac{1}{n^2} \times (n^7 + 9n^3 - 3) = -n^5 - 9n - \frac{3}{n^2}$
The simplified expression is:
$\boxed{-n^5 - 9n - \frac{3}{n^2}}$

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QuestionDivide and simplify. $\frac{3 n^{9}+27 n^{5}-9 n^{2}}{-3 n^{4}}$ $\frac{3 n^{9}+27 n^{5}-9 n^{2}}{-3 n^{4}}=$ $\square$ (Simplify your answer.)

Studdy Solution

STEP 1

1. The expression is a rational expression that needs to be simplified.
2. The numerator and the denominator can be simplified by factoring out common terms.
3. The division of polynomials involves simplifying each term separately.

STEP 2

1. Factor out the greatest common factor (GCF) from the numerator.
2. Simplify the division by dividing each term by the denominator.
3. Simplify the resulting expression.

STEP 3

Factor out the greatest common factor from the numerator $3n^9 + 27n^5 - 9n^2$ .
The GCF of the terms in the numerator is $3n^2$ .
$3n^9 + 27n^5 - 9n^2 = 3n^2(n^7 + 9n^3 - 3)$

STEP 4

Divide each term in the factored numerator by the denominator $-3n^4$ .
$\frac{3n^2(n^7 + 9n^3 - 3)}{-3n^4} = \frac{3n^2}{-3n^4} \times (n^7 + 9n^3 - 3)$

STEP 5

Simplify the fraction $\frac{3n^2}{-3n^4}$ .
$\frac{3n^2}{-3n^4} = -\frac{1}{n^2}$

STEP 6

Multiply the simplified fraction by the remaining polynomial.
$-\frac{1}{n^2} \times (n^7 + 9n^3 - 3) = -n^5 - 9n - \frac{3}{n^2}$
The simplified expression is:
$\boxed{-n^5 - 9n - \frac{3}{n^2}}$

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QuestionDivide and simplify. 3n9+27n5−9n2−3n4\frac{3 n^{9}+27 n^{5}-9 n^{2}}{-3 n^{4}}−3n43n9+27n5−9n2​ 3n9+27n5−9n2−3n4=\frac{3 n^{9}+27 n^{5}-9 n^{2}}{-3 n^{4}}=−3n43n9+27n5−9n2​= □\square□ (Simplify your answer.)

Studdy Solution

STEP 1

1. The expression is a rational expression that needs to be simplified.2. The numerator and the denominator can be simplified by factoring out common terms.3. The division of polynomials involves simplifying each term separately.

STEP 2

1. Factor out the greatest common factor (GCF) from the numerator.2. Simplify the division by dividing each term by the denominator.3. Simplify the resulting expression.

STEP 3

Factor out the greatest common factor from the numerator 3n9+27n5−9n23n^9 + 27n^5 - 9n^23n9+27n5−9n2.The GCF of the terms in the numerator is 3n23n^23n2.3n9+27n5−9n2=3n2(n7+9n3−3) 3n^9 + 27n^5 - 9n^2 = 3n^2(n^7 + 9n^3 - 3) 3n9+27n5−9n2=3n2(n7+9n3−3)

STEP 4

Divide each term in the factored numerator by the denominator −3n4-3n^4−3n4.3n2(n7+9n3−3)−3n4=3n2−3n4×(n7+9n3−3) \frac{3n^2(n^7 + 9n^3 - 3)}{-3n^4} = \frac{3n^2}{-3n^4} \times (n^7 + 9n^3 - 3) −3n43n2(n7+9n3−3)​=−3n43n2​×(n7+9n3−3)

STEP 5

Simplify the fraction 3n2−3n4\frac{3n^2}{-3n^4}−3n43n2​.3n2−3n4=−1n2 \frac{3n^2}{-3n^4} = -\frac{1}{n^2} −3n43n2​=−n21​

STEP 6

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QuestionDivide and simplify. $\frac{3 n^{9}+27 n^{5}-9 n^{2}}{-3 n^{4}}$ $\frac{3 n^{9}+27 n^{5}-9 n^{2}}{-3 n^{4}}=$ $\square$ (Simplify your answer.)

1. The expression is a rational expression that needs to be simplified.
2. The numerator and the denominator can be simplified by factoring out common terms.
3. The division of polynomials involves simplifying each term separately.

1. Factor out the greatest common factor (GCF) from the numerator.
2. Simplify the division by dividing each term by the denominator.
3. Simplify the resulting expression.

Factor out the greatest common factor from the numerator $3n^9 + 27n^5 - 9n^2$ .
The GCF of the terms in the numerator is $3n^2$ .
$3n^9 + 27n^5 - 9n^2 = 3n^2(n^7 + 9n^3 - 3)$

Divide each term in the factored numerator by the denominator $-3n^4$ .
$\frac{3n^2(n^7 + 9n^3 - 3)}{-3n^4} = \frac{3n^2}{-3n^4} \times (n^7 + 9n^3 - 3)$

Simplify the fraction $\frac{3n^2}{-3n^4}$ .
$\frac{3n^2}{-3n^4} = -\frac{1}{n^2}$