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+5x-7)divide(x-2)`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+5x-7` $$\mathsf{Divisor}$$ `=` `x-2` `=` 
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``(x-2)` `=` `x^2-2x` 
Subtract `x^2-2x` and write the difference one line below
Drop down `-7` and repeat the process to get the second term of the quotient
Second 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``(x-2)` `=` `7x-14` 
Subtract `7x-14` and write the difference one line below
Since `7` is not divisible by the divisor (`x-2`) 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+5x-7` $$\mathsf{Divisor}$$ `=` `x-2` $$\mathsf{Quotient}$$ `=` `x+7` $$\mathsf{Remainder}$$ `=` `7` $$\mathsf{\frac{P}{Divisor}}$$ `=` $$\mathsf{Quotient+\frac{R}{Divisor}}$$ Division of Polynomials $$\frac{x^2+5x-7}{x-2}$$ `=` $$x+7+\frac{7}{x-2}$$ Substitute `x+7+7/(x-2)` -
Question 3 of 5
3. Question
Solve`(2x^3+x^2+4x+1)/(x-3)`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` $$\mathsf{Divisor}$$ `=` `x-3` `=` 
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``(x-3)` `=` `2x^3-6x^2` 
Subtract `2x^3-6x^2` and write the difference one line below
Drop down `4x` and repeat the process to get the second term of the quotient
Second 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``(x-3)` `=` `7x^2-21x` 
Subtract `7x^2-21x` and write the difference one line below
Drop down `1` and repeat the process to get the second term of the quotient
Third 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``(x-3)` `=` `25x-75` 
Subtract `25x-75` and write the difference one line below
Since `76` is not divisible by the divisor (`x-3`) 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` $$\mathsf{Divisor}$$ `=` `x-3` $$\mathsf{Quotient}$$ `=` `2x^2+7x+25` $$\mathsf{Remainder}$$ `=` `76` $$\mathsf{\frac{P}{Divisor}}$$ `=` $$\mathsf{Quotient+\frac{R}{Divisor}}$$ Division of Polynomials $$\frac{2x^3+x^2+4x+1}{x-3}$$ `=` $$2x^2+7x+25+\frac{76}{x-3}$$ Substitute `2x^2+7x+25+76/(x-3)` -
Question 4 of 5
4. Question
Solve and give your answer in the format $$\mathsf{P=Divisor\times{Quotient}+R}$$`(8-y^2)divide(y-3)`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) `=` `8-y^2` `=` `-y^2+``0y` `+8` $$\mathsf{Divisor}$$ `=` `y-3` `=` 
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``(y-3)` `=` `-y^2+3y` 
Subtract `-y^2+3y` and write the difference one line below
Drop down `8` and repeat the process to get the second term of the quotient
Second 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``(y-3)` `=` `-3y+9` 
Subtract `-3y+9` and write the difference one line below
Since `-1` is not divisible by the divisor (`y-3`) 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) `=` `8-y^2` $$\mathsf{Divisor}$$ `=` `y-3` $$\mathsf{Quotient}$$ `=` `-y-3` $$\mathsf{Remainder}$$ `=` `-1` $$\mathsf{P}$$ `=` $$\mathsf{Divisor\times{Quotient}+R}$$ Division of Polynomials `=` $$(y-3)(-y-3)-1$$ Substitute `(y-3)(-y-3)-1` -
Question 5 of 5
5. Question
Solve`(x^4-x^3-17x^2-13x+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^4-x^3-17x^2-13x+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 below
Drop down `-17x^2` and repeat the process to get the second term of the quotient
Second 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^3-6x^2` 
Subtract `-3x^3-6x^2` and write the difference one line below
Drop down `-13x` and repeat the process to get the third term of the quotient
Third 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^2-22x` 
Subtract `-11x^2-22x` and write the difference one line below
Drop down `1` to see if a fourth term can be added to the quotient
Since `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^4-x^3-17x^2-13x+1` $$\mathsf{Divisor}$$ `=` `x^2+2x` $$\mathsf{Quotient}$$ `=` `x^2-3x-11` $$\mathsf{Remainder}$$ `=` `9x+1` $$\mathsf{\frac{P}{Divisor}}$$ `=` $$\mathsf{Quotient+\frac{R}{Divisor}}$$ Division of Polynomials $$\frac{x^4-x^3-17x^2-13x+1}{x^2+2x}$$ `=` $$x^2-3x-11+\frac{9x+1}{x^2+2x}$$ Substitute `x^2-3x-11+(9x+1)/(x^2+2x)`