Division of Polynomials
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Question 1 of 5
1. Question
Solve`(x^2+12x+7)divide6x`Hint
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A polynomial divided by a monomial can be turned into a fraction and then simplified.Transform the division into a fraction and then simplify`(x^2+12x+7)divide6x` `=` `(x^2+12x+7)/(6x)` `=` `(x^2)/(6x)+(12x)/(6x)+7/(6x)` Separate terms `=` `x/6+2+7/(6x)` Simplify `x/6+2+7/(6x)` 
Question 2 of 5
2. Question
Solve`(x^2+5x7)divide(x2)`Hint
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Division of Polynomials
$$\mathsf{\frac{P}{Divisor}=Quotient+\frac{R}{Divisor}}$$where $$\mathsf{P}$$ is the Polynomial and $$\mathsf{R}$$ is the RemainderLong Division
Use long division when a polynomial is divided by a binomialFirst, substitute components into the Long Division formula$$\mathsf{P}$$(Polynomial) `=` `x^2+5x7` $$\mathsf{Divisor}$$ `=` `x2` `=` Next, solve for each term of the quotientFirst term of the quotient:Divide the first term of the Polynomial by the first term of the Divisor. Place this above the Polynomial`x^2dividex` `=` `x` Multiply `x` to the divisor. Place this under the Polynomial`x``(x2)` `=` `x^22x` Subtract `x^22x` and write the difference one line belowDrop down `7` and repeat the process to get the second term of the quotientSecond term of the quotient:Divide the first term of the bottom expression by the first term of the Divisor. Place this above the Polynomial`7xdividex` `=` `7` Multiply `7` to the divisor. Place this one line below`7``(x2)` `=` `7x14` Subtract `7x14` and write the difference one line belowSince `7` is not divisible by the divisor (`x2`) anymore, it is left as the RemainderThis also means that the expression at the very top is the QuotientFinally, combine and substitute the components into the Division of Polynomials formula$$\mathsf{P}$$(Polynomial) `=` `x^2+5x7` $$\mathsf{Divisor}$$ `=` `x2` $$\mathsf{Quotient}$$ `=` `x+7` $$\mathsf{Remainder}$$ `=` `7` $$\mathsf{\frac{P}{Divisor}}$$ `=` $$\mathsf{Quotient+\frac{R}{Divisor}}$$ Division of Polynomials $$\frac{x^2+5x7}{x2}$$ `=` $$x+7+\frac{7}{x2}$$ Substitute `x+7+7/(x2)` 
Question 3 of 5
3. Question
Solve`(2x^3+x^2+4x+1)/(x3)`Hint
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Division of Polynomials
$$\mathsf{\frac{P}{Divisor}=Quotient+\frac{R}{Divisor}}$$where $$\mathsf{P}$$ is the Polynomial and $$\mathsf{R}$$ is the RemainderLong Division
Use long division when a polynomial is divided by a binomialFirst, substitute components into the Long Division formula$$\mathsf{P}$$(Polynomial) `=` `2x^3+x^2+4x+1` $$\mathsf{Divisor}$$ `=` `x3` `=` Next, solve for each term of the quotientFirst term of the quotient:Divide the first term of the Polynomial by the first term of the Divisor. Place this above the Polynomial`2x^3dividex` `=` `2x^2` Multiply `2x^2` to the divisor. Place this under the Polynomial`2x^2``(x3)` `=` `2x^36x^2` Subtract `2x^36x^2` and write the difference one line belowDrop down `4x` and repeat the process to get the second term of the quotientSecond term of the quotient:Divide the first term of the bottom expression by the first term of the Divisor. Place this above the Polynomial`7x^2dividex` `=` `7x` Multiply `7x` to the divisor. Place this one line below`7x``(x3)` `=` `7x^221x` Subtract `7x^221x` and write the difference one line belowDrop down `1` and repeat the process to get the second term of the quotientThird term of the quotient:Divide the first term of the bottom expression by the first term of the Divisor. Place this above the Polynomial`25xdividex` `=` `25` Multiply `25` to the divisor. Place this one line below`25``(x3)` `=` `25x75` Subtract `25x75` and write the difference one line belowSince `76` is not divisible by the divisor (`x3`) anymore, it is left as the RemainderThis also means that the expression at the very top is the QuotientFinally, combine and substitute the components into the Division of Polynomials formula$$\mathsf{P}$$(Polynomial) `=` `2x^3+x^2+4x+1` $$\mathsf{Divisor}$$ `=` `x3` $$\mathsf{Quotient}$$ `=` `2x^2+7x+25` $$\mathsf{Remainder}$$ `=` `76` $$\mathsf{\frac{P}{Divisor}}$$ `=` $$\mathsf{Quotient+\frac{R}{Divisor}}$$ Division of Polynomials $$\frac{2x^3+x^2+4x+1}{x3}$$ `=` $$2x^2+7x+25+\frac{76}{x3}$$ Substitute `2x^2+7x+25+76/(x3)` 
Question 4 of 5
4. Question
Solve and give your answer in the format $$\mathsf{P=Divisor\times{Quotient}+R}$$`(8y^2)divide(y3)`Hint
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Division of Polynomials
$$\mathsf{P=Divisor\times{Quotient}+R}$$where $$\mathsf{P}$$ is the Polynomial and $$\mathsf{R}$$ is the RemainderLong Division
Use long division when a polynomial is divided by a binomialFirst, notice that the `y`term with the power of 1 is missing. Add this term to the polynomial and arrange the terms in descending powers before using long division$$\mathsf{P}$$(Polynomial) `=` `8y^2` `=` `y^2+``0y` `+8` $$\mathsf{Divisor}$$ `=` `y3` `=` Next, solve for each term of the quotientFirst term of the quotient:Divide the first term of the Polynomial by the first term of the Divisor. Place this above the Polynomial`y^2dividey` `=` `y` Multiply `y` to the divisor. Place this under the Polynomial`y``(y3)` `=` `y^2+3y` Subtract `y^2+3y` and write the difference one line belowDrop down `8` and repeat the process to get the second term of the quotientSecond term of the quotient:Divide the first term of the bottom expression by the first term of the Divisor. Place this above the Polynomial`3ydividey` `=` `3` Multiply `3` to the divisor. Place this one line below`3``(y3)` `=` `3y+9` Subtract `3y+9` and write the difference one line belowSince `1` is not divisible by the divisor (`y3`) anymore, it is left as the RemainderThis also means that the expression at the very top is the QuotientFinally, combine and substitute the components into the Division of Polynomials formula$$\mathsf{P}$$(Polynomial) `=` `8y^2` $$\mathsf{Divisor}$$ `=` `y3` $$\mathsf{Quotient}$$ `=` `y3` $$\mathsf{Remainder}$$ `=` `1` $$\mathsf{P}$$ `=` $$\mathsf{Divisor\times{Quotient}+R}$$ Division of Polynomials `=` $$(y3)(y3)1$$ Substitute `(y3)(y3)1` 
Question 5 of 5
5. Question
Solve`(x^4x^317x^213x+1)/(x^2+2x)`Hint
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Division of Polynomials
$$\mathsf{\frac{P}{Divisor}=Quotient+\frac{R}{Divisor}}$$where $$\mathsf{P}$$ is the Polynomial and $$\mathsf{R}$$ is the RemainderLong Division
Use long division when a polynomial is divided by a binomialFirst, substitute components into the Long Division formula$$\mathsf{P}$$(Polynomial) `=` `x^4x^317x^213x+1` $$\mathsf{Divisor}$$ `=` `x^2+2x` `=` Next, solve for each term of the quotientFirst term of the quotient:Divide the first term of the Polynomial by the first term of the Divisor. Place this above the Polynomial`x^4dividex^2` `=` `x^2` Multiply `x^2` to the divisor. Place this under the Polynomial`x^2``(x^2+2x)` `=` `x^4+2x^3` Subtract `x^4+2x^3` and write the difference one line belowDrop down `17x^2` and repeat the process to get the second term of the quotientSecond term of the quotient:Divide the first term of the bottom expression by the first term of the Divisor. Place this above the Polynomial`3x^3dividex^2` `=` `3x` Multiply `3x` to the divisor. Place this one line below`3x``(x^2+2x)` `=` `3x^36x^2` Subtract `3x^36x^2` and write the difference one line belowDrop down `13x` and repeat the process to get the third term of the quotientThird term of the quotient:Divide the first term of the bottom expression by the first term of the Divisor. Place this above the Polynomial`11x^2dividex^2` `=` `11` Multiply `11` to the divisor. Place this one line below`11``(x^2+2x)` `=` `11x^222x` Subtract `11x^222x` and write the difference one line belowDrop down `1` to see if a fourth term can be added to the quotientSince `9x+1` is not divisible by the divisor (`x^2+2x`) anymore, it is left as the RemainderThis also means that the expression at the very top is the QuotientFinally, combine and substitute the components into the Division of Polynomials formula$$\mathsf{P}$$(Polynomial) `=` `x^4x^317x^213x+1` $$\mathsf{Divisor}$$ `=` `x^2+2x` $$\mathsf{Quotient}$$ `=` `x^23x11` $$\mathsf{Remainder}$$ `=` `9x+1` $$\mathsf{\frac{P}{Divisor}}$$ `=` $$\mathsf{Quotient+\frac{R}{Divisor}}$$ Division of Polynomials $$\frac{x^4x^317x^213x+1}{x^2+2x}$$ `=` $$x^23x11+\frac{9x+1}{x^2+2x}$$ Substitute `x^23x11+(9x+1)/(x^2+2x)`