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Question Number 194853 by horsebrand11 last updated on 17/Jul/23

Answered by cortano12 last updated on 17/Jul/23

$$\underbrace{⋐}$$

Commented by horsebrand11 last updated on 17/Jul/23

$$\underbrace{}$$

Answered by Rasheed.Sindhi last updated on 17/Jul/23

$$LetP\left(x\right)=8x^3+16x^2+kx+l$$$$=(2x-3)(4x^2+ax+b)$$$$8x^3+16x^2+kx+l=8x^3+(2a-12)x^2+(2b-3a)x-3b$$$$Comparingcoefficints:$$$$2a-12=16\Rightarrow a=14$$$$2b-3a=k\Rightarrow k=2b-42$$$$-3b=l\Rightarrow b=-\frac{l}{3}$$$$k=2(-\frac{l}{3})-42$$$$3k+2l=-126................\left(i\right)$$$$Similarly,$$$$letP\left(x\right)=8x^3+16x^2+kx+l=(2x+1)(4x^2+cx+d)+12$$$$8x^3+16x^2+kx+l$$$$=8x^3+(2c+4)x^2+(2d+c)x+d+12$$$$Comparingcoefficients:$$$$2c+4=16\Rightarrow c=6$$$$2d+c=k\Rightarrow 2d+6=k$$$$d+12=l\Rightarrow d=l-12$$$$2d+6=k\Rightarrow 2(l-12)+6=k$$$$k-2l=-18.................\left(ii\right)$$$$\left(i\right)+\left(ii\right):4k=-144\Rightarrow k=-36$$$$\left(i\right)\Rightarrow -36-2l=-18\Rightarrow l=-9$$$$$$$$$$

Commented by horsebrand11 last updated on 17/Jul/23

Answered by Rasheed.Sindhi last updated on 17/Jul/23

$$\underset{―}{\mathcal{B}ysyntheticdivision}$$$$\overline{)\begin{array}{ccccc}-\frac{1}{2}& 8& 16& k& l\\ & & -4& -6& -\frac{k}{2}+3\\ & 8& 12& k-6& l-\frac{k}{2}+3=12\end{array}}$$$$2l-k=18........\left(i\right)$$$$\overline{)\begin{array}{ccccc}\frac{3}{2}& 8& 16& k& l\\ & & 12& 42& \frac{3k}{2}+63\\ & 8& 28& k+42& l+\frac{3k}{2}+63=0\end{array}}$$$$2l+3k=-126........\left(ii\right)$$$$\left(ii\right)-\left(i\right):4k=-144\Rightarrow k=-36$$$$2l-(-36)=18\Rightarrow l=-9$$

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