An algebraic expression can be reduced by dividing its terms.
To divide algebraic expressions, it is essential to master operations on exponents and their properties.
When dividing algebraic expressions, several situations can arise.
All terms, whether they are like or not, can be divided. However, only like terms can be added or subtracted together.
Rarely does an equation consist of division only. When the situation arises, it is necessary to respect the order of operations to reduce the algebraic expression.
When dividing a monomial by a constant term, we divide the coefficient by the constant term.
||\begin{align} 12xy^{2}\div{3} &= \dfrac{12xy^{2}}{3} \\ &= \dfrac{12}{3}xy^{2} \\ &= 4xy^2 \end{align}||
When dividing a monomial by a monomial:
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We divide the coefficients.
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We subtract the exponents with the same bases.
The 2nd step is in fact the application of the properties of exponents on the quotient of powers of the same base.
||\begin{align} \dfrac{4^5}{4^3} &= \dfrac{\cancel{4^1} \times \cancel{4^1} \times \cancel{4^1} \times 4^1 \times 4^1}{\cancel{4^1}\times \cancel{4^1}\times \cancel{4^1}} \\ &=4^1 \times 4^1 = 4^2 = 4^{5-3} \\\\ \dfrac{x^6}{x^2} &= \dfrac{\cancel{x^1} \times \cancel{x^1} \times x^1 \times x^1 \times x^1 \times x^1}{\cancel{x^1}\times \cancel{x^1}} \\ &=x^1 \times x^1 \times x^1 \times x^1 = x^4 = x^{6-2} \end{align}||
||\begin{align} \dfrac{x^{3}y^{4}}{xy^{2}} &= x^{3-1}y^{4-2} \\ &= x^{2}y^{2} \end{align}||
||\begin{align} 25x^{3}y^{9}z\div5x^{3}y^{6} &= \dfrac{25x^{3}y^{9}z}{5x^{3}y^{6}}\\ \\ &= \dfrac{25}{5}x^{3-3}y^{9-6}z\\ \\ & = 5y^{3}z\end{align}||
To divide a polynomial by a monomial, we must:
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Distribute the division over each of the polynomial’s terms.
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For each term, divide the coefficients.
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Perform the following algebraic division: |
||\dfrac{18x^{2} + 54xy-6y+2}{6}|| |
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||\begin{align}= &\dfrac{18x^{2}}{6} + \dfrac{54xy}{6} - \dfrac{6y}{6} + \dfrac{2}{6} \end{align}|| |
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||\begin{align} = &\dfrac{18}{6}x^{2} + \dfrac{54}{6}xy - \dfrac{6}{6}y + \dfrac{2}{6} \\ = &\ \ \ 3\ x^2 +\ \ 9\ xy\ -\ 1\, y + \dfrac{1}{3} \end{align}|| |
Answer: Rewrite the expression by removing the coefficient |1.| Therefore, the answer is |3x^2 + 9xy - y + \dfrac{1}{3}.|
To divide a polynomial by a monomial, we must:
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Distribute the division over each of the polynomial’s terms.
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For each term, divide the coefficients.
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For each term, apply the property of the exponents on the quotient of powers with the same base by subtracting the exponents of the same base.
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Perform the following algebraic division: |
|\quad\ \ \ \dfrac {12xy^{2} + 6x^{8}y^{6}}{-3x^{3}y^{4}}| |
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|\begin{align} &= \dfrac {12xy^{2}}{-3x^{3}y^{4}} + \dfrac {6x^{8}y^{6}}{-3x^{3}y^{4}} \end{align}| |
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|\begin{align} &=\dfrac{12}{-3}\dfrac{xy^2}{x^3y^4} + \dfrac{6}{-3}\dfrac{x^8y^6}{x^3y^4} \\ &= \left( -4 \dfrac{xy^2}{x^3y^4}\right) + \left( -2 \dfrac{x^8y^6}{x^3y^4}\right) \end{align}| |
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|\begin{align} &=-4\dfrac {xy^{2}}{x^{3}y^{4}} -2\dfrac {x^{8}y^{6}}{x^{3}y^{4}} \\\\ &= -4x^{1-3}y^{2-4} -2x^{8-3}y^{6-4}\\\\ &= -4x^{-2}y^{-2} - 2x^{5}y^{2}\end{align}| |
Answer: Rewrite the answer so that there is no negative exponent. Therefore, the answer is |\dfrac {-4}{x^{2}y^{2}} - 2x^{5}y^{2}.|
Never leave negative exponents in the solution. Use a property of exponents to make them positive. If they are in the denominator, they go to the numerator and vice versa.
It is written as follows: |a^{-n} = \dfrac{1}{a^n}| et |\dfrac{1}{a^{-n}} = a^{n}.|