Remainder Theorem
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Question 1 of 4
1. Question
Write “F” if the binomial is a Factor and “N” if it is not a Factor of the polynomial:`P(x)=x^3+2x^25x6`
`(a) (x2)=` (F, f)`(b) (2x1)=` (N, n)`(c) (x+3)=` (F, f)
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Remainder Theorem
If `(P(x))/(xa)` then `P(a)=r`where `r` is the RemainderNote that if `r=0`, then `(xa)` is a Factor of `P(x)``(a)` Identify if `(x2)` is a factor of `P(x)=x^3+2x^25x6`.First, label the variable `a` by equating the binomial to `0` and solving for `x``x2` `=` `0` `x2` `+2` `=` `0` `+2` Add `2` to both sides `x` `=` `2` Hence, `a=2`Next, substitute `a` into the polynomial to solve for `r``P(x)=x^3+2x^25x6``a=2``P(a)` `=` `r` Remainder Theorem `P(a)` `=` `a^3+2a^25a6` Replace `x` with `a` `P(2)` `=` `2^3+2(2)^25(2)6` Substitute `a=2` `=` `8+2(4)106` `=` `8+8106` `=` `1616` `r` `=` `0` Since `r=0`, we can say that `(x2)` is a factor of `P(x)=x^3+2x^25x6``(b)` Identify if `(2x1)` is a factor of `P(x)=x^3+2x^25x6`.First, label the variable `a` by equating the binomial to `0` and solving for `x``2x1` `=` `0` `2x1` `+1` `=` `0` `+1` Add `1` to both sides `2x` `=` `1` `2x``:2` `=` `1``:2` Divide both sides by `2` `x` `=` `1/2` Hence, `a=1/2`Next, substitute `a` into the polynomial to solve for `r``P(x)=x^3+2x^25x6``a=1/2``P(a)` `=` `r` Remainder Theorem `P(a)` `=` `a^3+2a^25a6` Replace `x` with `a` `P(1/2)` `=` `(1/2)^3+2(1/2)^25(1/2)6` Substitute `a=1/2` `=` `1/8+2(1/4)5/26` `=` `1/8+4/820/848/8` `r` `=` `(63)/8` Since $$r\neq0$$, we can say that $$(2x1)$$ is not a factor of $$P(x)=x^3+2x^25x6$$`(c)` Identify if `(x+3)` is a factor of `P(x)=x^3+2x^25x6`.First, label the variable `a` by equating the binomial to `0` and solving for `x``x+3` `=` `0` `x+3` `3` `=` `0` `3` Subtract `3` from both sides `x` `=` `3` Hence, `a=3`Next, substitute `a` into the polynomial to solve for `r``P(x)=x^3+2x^25x6``a=3``P(a)` `=` `r` Remainder Theorem `P(a)` `=` `a^3+2a^25a6` Replace `x` with `a` `P(3)` `=` `(3)^3+2(3)^25(3)6` Substitute `a=3` `=` `27+2(9)+156` `=` `27+18+156` `=` `33+33` `r` `=` `0` Since `r=0`, we can say that `(x+3)` is a factor of `P(x)=x^3+2x^25x6``(a)` Factor`(b)` Not a factor`(c)` Factor 

Question 2 of 4
2. Question
Solve for the remainder of:`(8x+x^2+3)divide(3+x)` `r=` (12)
Hint
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Well Done!
Incorrect
Remainder Theorem
If `(P(x))/(xa)` then `P(a)=r`where `r` is the RemainderLong Division
Use long division when a polynomial is divided by a binomialNote that if `r=0`, then `(xa)` is a Factor of `P(x)`Method OneFind the remainder using long divisionFirst, arrange the terms in descending order of powers.$$\mathsf{P}$$(Polynomial) `=` `8x+x^2+3` `=` `x^2+8x+3` $$\mathsf{Divisor}$$ `=` `3+x` `=` `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`x^2dividex` `=` `x` Multiply `x` to the divisor. Place this under the Polynomial`x``(x+3)` `=` `x^2+3x` Subtract `x^2+3x` and write the difference one line belowDrop down `3` 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`5xdividex` `=` `5` Multiply `5` to the divisor. Place this under the Polynomial`5``(x+3)` `=` `5x+15` Subtract `5x+15` and write the difference one line below`12` is not divisible by the divisor (`x+3`) anymore, hence `r=12``r=12`Method TwoFind the remainder using remainder theoremFirst, label the variable `a` by equating the divisor to `0` and solving for `x``x+3` `=` `0` `x+3` `3` `=` `0` `3` Subtract `3` from both sides `x` `=` `3` Hence, `a=3`Next, substitute `a` into the polynomial to solve for `r``P(x)=8x+x^2+3``a=3``P(a)` `=` `r` Remainder Theorem `P(a)` `=` `8a+a^2+3` Replace `x` with `a` `P(3)` `=` `8(3)+(3)^2+3` Substitute `a=3` `=` `24+9+3` `r` `=` `12` `r=12` 
Question 3 of 4
3. Question
Solve for the remainder of:`(x^43x^3+2x+8)/(x2)` `r=` (4)
Hint
Help VideoCorrect
Excellent!
Incorrect
Remainder Theorem
If `(P(x))/(xa)` then `P(a)=r`where `r` is the RemainderLong Division
Use long division when a polynomial is divided by a binomialNote that if `r=0`, then `(xa)` is a Factor of `P(x)`Method OneFind the remainder using long divisionFirst, notice that the term with the power of `2` is missing. Add this term to the polynomial before using long division$$\mathsf{P}$$(Polynomial) `=` `x^43x^3+2x+8` `=` `x^43x^3+``0x^2``+2x+8` $$\mathsf{Divisor}$$ `=` `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^4dividex` `=` `x^3` Multiply `x^3` to the divisor. Place this under the Polynomial`x^3``(x2)` `=` `x^42x^3` Subtract `x^42x^3` and write the difference one line belowDrop down `0x^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`x^3dividex` `=` `x^2` Multiply `x^2` to the divisor. Place this one line below`x^2``(x2)` `=` `x^3+2x^2` Subtract `x^3+2x^2` and write the difference one line belowDrop down `2x` 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`2x^2dividex` `=` `2x` Multiply `2x` to the divisor. Place this under the Polynomial`2x``(x2)` `=` `2x^2+4x` Subtract `2x^2+4x` and write the difference one line belowDrop down `8` and repeat the process to get the fourth term of the quotientFourth term of the quotient:Divide the first term of the bottom expression by the first term of the Divisor. Place this above the Polynomial`2xdividex` `=` `2` Multiply `2` to the divisor. Place this under the Polynomial`2``(x2)` `=` `2x+4` Subtract `2x+4` and write the difference one line below`4` is not divisible by the divisor (`x2`) anymore, hence `r=4``r=4`Method TwoFind the remainder using remainder theoremFirst, label the variable `a` by equating the divisor to `0` and solving for `x``x2` `=` `0` `x2` `+2` `=` `0` `+2` Add `2` to both sides `x` `=` `2` Hence, `a=2`Next, substitute `a` into the polynomial to solve for `r``P(x)=x^43x^3+2x+8``a=2``P(a)` `=` `r` Remainder Theorem `P(a)` `=` `a^43a^3+2a+8` Replace `x` with `a` `P(2)` `=` `2^43(2)^3+2(2)+8` Substitute `a=2` `=` `163(8)+4+8` `=` `1624+4+8` `r` `=` `4` `r=4` 
Question 4 of 4
4. Question
Solve for the remainder of:`(a) (x^45x^3+3x^2+8)/(x2)``(b) (3x^4x^2+2x+6)/(x+1)`
`(a) r=` (4)`(b) r=` (6)
Hint
Help VideoCorrect
Great Work!
Incorrect
Remainder Theorem
If `(P(x))/(xa)` then `P(a)=r`where `r` is the Remainder`(a)` Solve for the remainder of `(x^45x^3+3x^2+8)/(x2)`.First, label the variable `a` by equating the divisor to `0` and solving for `x``x2` `=` `0` `x2` `+2` `=` `0` `+2` Add `2` to both sides `x` `=` `2` Hence, `a=2`Next, substitute `a` into the polynomial to solve for `r``P(x)=x^45x^3+3x^2+8``a=2``P(a)` `=` `r` Remainder Theorem `P(a)` `=` `a^45a^3+3a^2+8` Replace `x` with `a` `P(2)` `=` `(2)^45(2)^3+3(2)^2+8` Substitute `a=2` `=` `165(8)+3(4)+8` `=` `1640+12+8` `r` `=` `4` `(b)` Solve for the remainder of `(3x^4x^2+2x+6)/(x+1)`.First, label the variable `a` by equating the divisor to `0` and solving for `x``x+1` `=` `0` `x+1` `1` `=` `0` `1` Subtract `1` from both sides `x` `=` `1` Hence, `a=1`Next, substitute `a` into the polynomial to solve for `r``P(x)=3x^4x^2+2x+6``a=1``P(a)` `=` `r` Remainder Theorem `P(a)` `=` `3a^4a^2+2a+6` Replace `x` with `a` `P(1)` `=` `3(1)^4(1)^2+2(1)+6` Substitute `a=1` `=` `3(1)12+6` `=` `312+6` `r` `=` `6` `(a)4``(b) 6` 