Algebra 1 : How to divide monomial quotients

Study concepts, example questions & explanations for Algebra 1

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Example Questions

Example Question #1 : How To Divide Monomial Quotients

Evaluate

\(\displaystyle \frac{(12w^{2}-15w+21w^{3})}{3w}\)

Possible Answers:

\(\displaystyle -1+7w^{2}\)

 \(\displaystyle 7w^{2}+4w-5\)

\(\displaystyle 2w^{2}+4w-5\)

\(\displaystyle 2w^{2}-4w+5\)

Correct answer:

 \(\displaystyle 7w^{2}+4w-5\)

Explanation:

\(\displaystyle \frac{(12w^{2}-15w+21w^{3})}{3w}\)

When dividing a polynomial by a monomial, we can use a divison called, term-by-term, dividing each of the top terms by the monomial.

\(\displaystyle \frac{12w^{2}}{3w}-\frac{15w}{3w}+\frac{21w^{3}}{3w}\)

Simplify.

\(\displaystyle 4w-5+7w^{2}\)

Rewrite it with the leading coefficent first, 

Final Answer: \(\displaystyle 7w^{2}+4w-5\)

Example Question #461 : Variables

Simplify the fraction to its lowest terms:

\(\displaystyle \frac{18x^4y^3z}{6xy^2z^3}\)

Possible Answers:

\(\displaystyle 3x^3yz^2\)

\(\displaystyle 3x^3yz^2\)

\(\displaystyle 12x^3yz^2\)

\(\displaystyle 3x^5y^5z^4\)

\(\displaystyle 3x^3yz^{-2}\)

Correct answer:

\(\displaystyle 3x^3yz^{-2}\)

Explanation:

The first step is to divide the constants, 18 and 6, by the LCM, 6, to get 3. When dividing variables, if the variable is present in the numerator and denominator, subtract the exponent found in the numerator by the exponent in the denominator.

For \(\displaystyle x\), you have \(\displaystyle 4-1=3\).

For \(\displaystyle y\), you have \(\displaystyle 3-2=1\).

For \(\displaystyle z\), you have \(\displaystyle 1-3=-2\).

Then write the simplified answer as one term:

\(\displaystyle 3x^3yz^{-2}\)

Example Question #3 : How To Divide Monomial Quotients

Divide:

\(\displaystyle \frac{9x^4+6x^3-12x}{3x}\)

Possible Answers:

\(\displaystyle 3x^3+2x^2-4\)

\(\displaystyle 3x^3+2x^2+4\)

\(\displaystyle 6x^3+3x^2-9\)

\(\displaystyle 6x^3+3x^2+9\)

Correct answer:

\(\displaystyle 3x^3+2x^2-4\)

Explanation:

\(\displaystyle \frac{9x^4+6x^3-12x}{3x}=\frac{9x^4}{3x}+\frac{6x^3}{3x}-\frac{12x}{3x}=3x^3+2x^2-4\)

Example Question #41 : Monomials

Siimplify:

\(\displaystyle \frac{2x^{2}y^{3}z - 6xy^{2}z^{2}+10x^{3}yz^{3}}{2xyz} =\ ?\)

Possible Answers:

\(\displaystyle xyz-3yz+5x^{2}yz^{2}\)

\(\displaystyle xy^{2}-3xz+5xy\)

\(\displaystyle x^{2}y - 3xyz + 5x^{2}yz^{2}\)

\(\displaystyle xy^{2}-3yz+5x^{2}z^{2}\)

\(\displaystyle xy - yz+5xz^{2}\)

Correct answer:

\(\displaystyle xy^{2}-3yz+5x^{2}z^{2}\)

Explanation:

For any polynomial division, divide each term in the numerator individually by the denominator:

\(\displaystyle \frac{2x^{2}y^{3}z - 6xy^{2}z^{2}+10x^{3}yz^{3}}{2xyz} =\) \(\displaystyle \frac{2x^{2}y^{3}z}{2xyz} - \frac{6xy^{2}z^{2}}{2xyz} + \frac{10x^{3}yz^{3}}{2xyz}=\)

 

\(\displaystyle xy^{2}-3yz+5x^{2}z^{2}\)

Example Question #2 : How To Divide Monomial Quotients

Simplify:

\(\displaystyle \frac{14x^2y}{7x}\)

Possible Answers:

\(\displaystyle 98xy\)

\(\displaystyle 98x^3y\)

\(\displaystyle \frac{1}{2xy}\)

\(\displaystyle 2xy\)

\(\displaystyle 2y\)

Correct answer:

\(\displaystyle 2xy\)

Explanation:

When dividing monomials, consider the coefficients and variables separately. Rewrite the expression as  \(\displaystyle \left(\frac{14}{7}\right)\left(\frac{x^2}{x}\right)\left(y\right)\), grouping common bases. For the coeffiecients, we can divide normally: \(\displaystyle \left(\frac{14}{7} \right )=2\). For the variables, we can keep the common base and subtract the exponents: \(\displaystyle \left(\frac{x^2}{x} \right )=x^{2-1}=x\). Then, multiply each portion all back together to obtain \(\displaystyle 2xy\).

Example Question #1 : How To Divide Monomial Quotients

Simplify the expression.

\(\displaystyle \small \frac{(x^2y^2)(x^3y^4z)}{x^4y^3z^4}\)

Possible Answers:

\(\displaystyle \small \frac{x^2y^5}{z^3}\)

\(\displaystyle \small \frac{z^3}{xy^3}\)

\(\displaystyle \small \frac{xy^3}{z^3}\)

\(\displaystyle \small \frac{z^3}{x^2y^5}\)

Correct answer:

\(\displaystyle \small \frac{xy^3}{z^3}\)

Explanation:

\(\displaystyle \small \frac{(x^2y^2)(x^3y^4z)}{x^4y^3z^4}\)

Because we are only multiplying terms in the numerator, we can disregard the parentheses.

\(\displaystyle \small \frac{(x^2y^2)(x^3y^4z)}{x^4y^3z^4}=\frac{x^2y^2x^3y^4z}{x^4y^3z^4}\)

To combine like terms in the numerator, we add their exponents.

\(\displaystyle \small \frac{x^5y^6z}{x^4y^3z^4}\)

To combine like terms between the numerator and denominator, subtract the denominator exponent from the numerator exponent.

\(\displaystyle x^{5-4}y^{6-3}z^{1-4}=xy^3z^{-3}\)

Remember that any negative exponents stay in the denominator.

\(\displaystyle \frac{xy^3}{z^3}\)

Example Question #2 : Simplifying Expressions

Simplify:        \(\displaystyle \frac{5p^{7}}{25p^{^{4}}}\)

Possible Answers:

\(\displaystyle \frac{p^{-3}}{5}\)

\(\displaystyle 5p^{3}\)

\(\displaystyle \frac{p^{7/4}}{5}\)

\(\displaystyle \frac{p^{3}}{5}\)

\(\displaystyle \frac{p}{5}\)

Correct answer:

\(\displaystyle \frac{p^{3}}{5}\)

Explanation:

\(\displaystyle p^{7}\) and \(\displaystyle p^{4}\) cancel out, leaving \(\displaystyle p^{3}\) in the numerator. 5 and 25 cancel out, leaving 5 in the denominator

Example Question #5 : How To Divide Monomial Quotients

Simplify the following:

\(\displaystyle \frac{4x+6x^2}{2x}\)

Possible Answers:

\(\displaystyle 2+3x\)

\(\displaystyle 2x(2-3x)\)

\(\displaystyle \frac{1}{x}-1\frac{1}{2}\)

\(\displaystyle \frac{2-3x}{2x}\)

\(\displaystyle \frac{2}{2x}-\frac{3}{2}\)

Correct answer:

\(\displaystyle 2+3x\)

Explanation:

\(\displaystyle \frac{4x+6x^2}{2x}\)

First, let us factor the numerator:

\(\displaystyle 4x+6x^2=x(4+6x)=2x(2+3x)\)

\(\displaystyle \frac{2x(2+3x)}{2x}=2+3x\)

Example Question #51 : Monomials

Simplify the following:

\(\displaystyle \frac{-2pl^2}{3yz}\div \frac{9lp^2}{5xy}\)

Possible Answers:

\(\displaystyle \frac{-18p^{3}l^{3}}{15xy^{2}z}\)

\(\displaystyle 1\)

\(\displaystyle \frac{-10lx}{27zp}\)

\(\displaystyle \frac{-18}{15xyz}\)

\(\displaystyle \frac{5}{9}\)

Correct answer:

\(\displaystyle \frac{-10lx}{27zp}\)

Explanation:

First, flip the numerator and the denominator of the second fraction to turn the division into multiplication.

\(\displaystyle \frac{-2pl^2}{3yz}\div \frac{9lp^2}{5xy}=\frac{-2pl^2}{3yz} * \frac{5xy}{9lp^2}=\frac{-2pl^25xy}{3yz9lp^2}\)

We can then cancel like terms.

From both the numerator and denominator, remove one \(\displaystyle p\), remove one \(\displaystyle l\), and remove one \(\displaystyle y\):

\(\displaystyle \frac{-2l5x}{3z9p}\)

Then we finish by multiplying the constants:

\(\displaystyle \frac{-10lx}{27zp}\)

Example Question #3 : How To Divide Monomial Quotients

Simplify this expression:

         \(\displaystyle \frac{x^{2}y^{-3}z^{4}}{x^{3}y^{-2}z^{2}}\)

Possible Answers:

\(\displaystyle xyz^{2}\)

\(\displaystyle \frac{1}{xyz^{2}}\)

\(\displaystyle \frac{xy}{z^{2}}\)

\(\displaystyle \frac{yz^{2}}{x}\)

\(\displaystyle \frac{z^{2}}{xy}\)

Correct answer:

\(\displaystyle \frac{z^{2}}{xy}\)

Explanation:

When different powers of the same variable are multiplied, the exponents are added.  When different powers of the same variable are divided, the exponents are subtracted.  So, as an example:

\(\displaystyle \frac{x^{3}}{x} = x^{\left ( 3-1\right )} = x^{2}\)

For the above problem, 

\(\displaystyle \frac{x^{2}}{x^{3}} = \frac{1}{x} ; \frac{y^{-3}}{y^{-2}} = \frac{y^{2}}{y^{3}} = \frac{1}{y} ; \frac{z^{4}}{z^{2}} = z^{2}\)

 

Therefore, the expression simplifies to:

\(\displaystyle \left ( \frac{1}{x} \right ) \left ( \frac{1}{y} \right ) z^{2} = \frac{z^{2}}{xy}\)

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