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This is the second of our many quizzes on simplifying polynomials. Click on this to access more such competitive quizzes on polynomials.

Simplifying Polynomials Quiz 2

mathematics

  If one zero of the polynomial $f(x)={{x}^{2}}-dx+200$ is twice of the other, then find the value of $d$, if $d > 0$.

 Take two zeros as $a$ and $2a$. Then, use the formula of sum and product of zeros to find the value of $d$.

 $\begin{align} & \text{Let “a” and “2a” are the two zeros of a polynomial “f(x)”. }\\ & \Rightarrow a+2a=d \text{ and } 2{{a}^{2}}=200 \\ & \Rightarrow 3a=d\text{ and }a=\pm 10 \\ & \therefore d=30\text{        }(\because d > 0)\end{align}$ 

 Factorize: ${{a}^{3}}-12{{b}^{3}}-4a{{b}^{2}}+3{{a}^{2}}b$.

 First we write $({{a}^{3}},4a{{b}^{2}})\text{ and }(12{{b}^{3}},3{{a}^{2}}b)$ together and then factorize to get the required answer.

 $\begin{align} & a^3-12b^3-4ab^2+3a^2b \\ &\Rightarrow a(a^2-4b^2)+3b(a^2-4b^2) \\ &\Rightarrow (a+3b)(a-2b)(a+2b) \end{align}$

 Find the value of $c$ and $d$, if roots of the polynomial ${{y}^{2}}+cy+d$ are six times the roots of polynomial $3{{x}^{2}}-2x-1$.

 First find the roots of the polynomial $3{{x}^{2}}-2x-1$ and then find the roots of the other polynomial. Then, use the formula of sum and product of roots to find the value of $c$ and $d$.

 $\begin{align} & \text{Given }3{{x}^{2}}-2x-1 \text{ can be written as}\\ & \Rightarrow 3{{x}^{2}}-3x+x-1 \Rightarrow (3x+1)(x-1) \\ & \text{The roots of the polynomial }3{{x}^{2}}-2x-1\text{ is }\\ & \Rightarrow x=1,\dfrac{-1}{3}\\ &\text{ As }{{y}^{2}}+cy+d\text{ is six times the roots of polynomial of }3{{x}^{2}}-2x-1\\ & \text{Then roots the roots are }\\ &\Rightarrow y=6 \left( 1,\dfrac{-1}{3}\right) = 6,-2\\ &\Rightarrow \text{ Sum of the roots }= 6 - 2 = - c\text{ and Product }= 6 \times -2 = d \\ &\Rightarrow c=-4 \text{ and } d=-12 \end{align}$

 The number of mobiles sold by a company in a year is given by $A(x)=200x-900$ and the cost of each mobile is given by $B(x)={{x}^{2}}-10$, where $x$ is the number of months in a year. Find the revenue generated by a company in a year.

 First multiply both the given polynomials and then put $x = 12$ to get the answer.

 $\begin{align} & \text{Revenue generated by a company in a year is }(200x-900)({{x}^{2}}-10) \\ &\Rightarrow 200{{x}^{3}}-2000x-900{{x}^{2}}+9000 \\ &\Rightarrow 100\left( 2{{(12)}^{3}}-20(12)-9{{(12)}^{2}}+90 \right) \,\,\,\,\,\,\,\,(\because x \text{is the number of months})\\ &\Rightarrow 201000 \end{align}$ 

 Find the speed of the car, if the distance covered by a car is given by $q(x)=2x(x-4)+10(x-3)-10$  and time taken by a car to cover the same distance is given by $r(x)=x+5$?

 First convert polynomial $q(x)$ in quadratic form and then factorize it to get the required answer.

 $\begin{align} & \text{Speed}=\dfrac{\text{Distance}}{\text{Time}} \\ &\Rightarrow Speed=\dfrac{2x(x-4)+10(x-3)-10}{x+5} \\ &\Rightarrow \dfrac{2({{x}^{2}}+x-20)}{x+5} \\ &\Rightarrow \dfrac{2(x+5)(x-4)}{x+5}=2(x-4) \end{align}$