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This is the second of our many quizzes on Synthetic division. Click on this to access more such competitive quizzes on Algebra.

Synthetic division Quiz 2

mathematics

 Evaluate using Synthetic division: $\dfrac{2{{x}^{3}}-14{{x}^{2}}-11x-21}{x-3}$.

 First write the coefficients of the dividend inside and take root of divisor i.e.  $x=3$ on the left side. Now, carry-down the first coefficient and multiply it by $3$ and then carry the result up into the next column, and so on.

  <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/a995f951-f5eb-411f-81bd-1d689e2784405433306935257024771.png"/> $ \\ \therefore \dfrac{f(x)}{g(x)}=2{{x}^{3}}+8x+13+\dfrac{18}{x-3}$

 Find the value of $a$, if the remainder of $\dfrac{f(x)}{g(x)}$ is $10$, by using Synthetic division, where $f(x)={{x}^{3}}-2{{x}^{2}}+9x+a$ and $g(x)=x-1$.

 First write the coefficients of the dividend inside and take root of divisor i.e.  $x=1$ on the left side. Now, carry-down the first coefficient and multiply it by $1$ and then carry the result up into the next column, and so on.

  <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/828b93a6-97d9-411b-a349-8b4a8cac1cf93772722662007863128.png"/> $ \\ $ The last carry-down value is the remainder. $ \therefore \dfrac{f(x)}{g(x)}={{x}^{2}}-x+8+\dfrac{8+a}{x-3} \\  \Rightarrow 8+a=10\Rightarrow a=2 \\ $

 Find the remainder of $\dfrac{f(x)}{g(x)}$ by using Synthetic division, where $f(x)={{x}^{4}}-2{{x}^{3}}+3{{x}^{2}}-2$ and $g(x)=\left( x-1 \right)\left( x-3 \right)$.

<img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/ac73d675-0efd-4087-9860-a21813425a222304577101357983546.png"/> $ \\ \therefore \dfrac{f(x)}{g(x)}=\dfrac{{{x}^{3}}-{{x}^{2}}+2x+2}{x-3} \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/0dcc5f13-212f-46e3-92ac-c8e031c92c526033079124959165703.png"/> $ \\ \therefore \dfrac{f(x)}{g(x)}={{x}^{2}}+2x+8+\dfrac{26}{x-3}$

 Find the difference between the remainders of $\dfrac{f(x)}{g(x)}$ and $\dfrac{h(x)}{i(x)}$ by using Synthetic division, where $f(x)={{x}^{4}}+8{{x}^{2}}-16$, $h(x)={{x}^{3}}-8x+21$, $i(x)=x-2$ and $g(x)=x-1$.

 First, write the coefficients inside an upside-down division symbol, put $x=1$ and $x=2$ on the left side, multiply the first carry-down value by $1\text{ and 2}$, and carry the result up into the next column, and so on.

 <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/19d2682f-76db-486a-ae33-4b0be47d52a14655017500875453375.png"/> $ \\ \therefore \dfrac{f(x)}{g(x)}={{x}^{3}}+{{x}^{2}}+9x+9-\dfrac{7}{x-1} \\ $ <sub></sub><img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/006f5ce1-bc0b-49ae-ab99-6753a0d8d7e32970866026024074110.png"/> $ \\ \dfrac{h(x)}{i(x)}={{x}^{2}}+2x-4+\dfrac{13}{x-2} \\ $ The last carry-down value is the remainder. $ \\ $ Therefore, the difference between both the remainders $=13-(-7)=13+7=20$

 Find the new polynomial formed by adding the quotients of $\dfrac{f(x)}{g(x)}$ and $\dfrac{h(x)}{g(x)}$ by using Synthetic division, where $f(x)={{x}^{5}}-7{{x}^{4}}+10{{x}^{2}}+20x-12$, $h(x)={{x}^{4}}-x$ and $g(x)=\left( x-2 \right)$.

 First write the coefficients of the dividend inside and take root of divisor i.e.  $x=2$ on the left side. Now, carry-down the first coefficient and multiply it by $2$ and then carry the result up into the next column, and so on.

 <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/5bdc9635-6801-4efc-b4c9-4e0c68d51b039022698823646187715.png"/> $ \\ \therefore \dfrac{f(x)}{g(x)}={{x}^{4}}-5{{x}^{3}}-10{{x}^{2}}-10x-\dfrac{12}{x-2} \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/30f4fbd9-6fd8-4301-9433-3cfdf516eadc7042245351312528623.png"/> $ \\ \dfrac{h(x)}{g(x)}={{x}^{3}}+2{{x}^{2}}+4x+7+\dfrac{14}{x-2} \\ $ Except the last carry-down number, the other numbers form the coefficients of the quotient. $ \\ $ Therefore, the new polynomial $={{x}^{4}}-5{{x}^{3}}-10{{x}^{2}}-10x+{{x}^{3}}+2{{x}^{2}}+4x+7={{x}^{4}}-4{{x}^{3}}-8{{x}^{2}}-6x+7$