Chapter 7

$$ \left.x^{3}-5 x^{2}+3 x+9 $$

$$ \left.2 x^{3}-18 x^{2}+108 $$

$$ \left.x^{3}+\frac{1}{2} x^{2}-\frac{1}{4} x-\frac{1}{8}, $$

$$ x^{4}-12 x^{3}+54 x^{2}-108 x+81 $$

$$ x^{4}-x^{3}-3 x^{2}+5 x-2 $$

$$ 2 x^{4}-3 x^{3}-x^{2}+3 x-1 $$

$$ x^{4}+14 x^{3}+71 x^{2}+154 x+120 . $$

$$ x^{4}-\frac{7}{3} x^{3}-11 x^{2}+\frac{49}{3} x+4 $$

$$ x^{4}-4 x^{3}+4 x^{2}-1 $$

$$ x^{5}-5 x^{4}+10 x^{3}-10 x^{2}+5 x-1 $$

$$ x^{5}-2 x^{4}-2 x^{3}+8 x^{2}-7 x+2 . $$

$$ 4 x^{5}-5 x^{4}-11 x^{3}+23 x^{2}-13 x+2 . $$

$$ 2 x^{5}-9 x^{4}+8 x^{3}+15 x^{2}-28 x+12 $$

$$ 3 x^{5}-19 x^{4}+9 x^{3}+71 x^{2}-84 x+20 $$

$$ x^{5}-7 x^{4}+16 x^{3}-16 x^{2}+7 x-1 $$

Show that the polynomial \(P(x)=x^{5}+x^{3}+2 x+1\) cannot have a positive real root.

Show that the polynomial \(P(x)=x^{7}+x^{5}-2 x^{4}+x^{3}-3 x^{2}+7 x-5\) cannot have a negative real root.

Find the nature of roots of the polynomial \(P(x)=x^{3}+x+1\).

Find the nature of roots of the polynomial \(P(x)=3 x^{4}+12 x^{2}+5 x-4\).

Find the nature of roots of the polynomial \(P(x)=2 x^{4}+5 x^{2}+3\).

Find the nature of roots of the polynomial \(P(x)=2 x^{8}+3 x^{4}+x^{2}+7\).

Find the nature of roots of the polynomial \(P(x)=x^{9}+2 x^{5}+3 x^{3}+x\).

Show that the polynomial \(P(x)=2 x^{7}-x^{4}+4 x^{3}-5\) has at least four imaginary roots.

Find the least possible number of imaginary roots of the polynomial \(P(x)=x^{9}-x^{5}+x^{4}+x^{2}+1\).

$$ \frac{3 x-2}{2 x-3}<3 $$

Show that the polynomial \(P(x)=x^{9}+5 x^{8}-x^{3}+7 x+2\) has at least four imaginary roots.

Show that the polynomial \(P(x)=x^{10}-4 x^{6}+x^{4}-2 x-3\) has at least four imaginary roots.

$$ x^{3}+4 x^{2}+6 x+3=0 $$

$$ (x-1)^{3}+(2 x+3)^{3}=27 x^{3}+8 $$

$$ 2 x^{4}-x^{3}-9 x^{2}+13 x-5=0 $$

$$ 8 x^{4}+6 x^{3}-13 x^{2}-x+3=0 $$

$$ x^{4}-4 x^{3}-19 x^{2}+106 x-120=0 $$

$$ (x-4)(x-5)(x-6)(x-7)=1680 $$

$$ (6 x+5)^{2}(3 x+2)(x+1)=35 $$

$$ x^{4}-2 x^{3}+x-132=0 $$

$$ (x-4)(x+2)(x+8)(x+14)=304 $$

$$ \left(x^{2}+x+1\right)\left(2 x^{2}+2 x+3\right)=3\left(1-x-x^{2}\right) $$

$$ \frac{x+1}{x+3}+\frac{4}{x+7}=1 $$

$$ \frac{6 x-5}{4 x-3}=\frac{3 x+3}{2 x+5} $$

$$ \frac{1}{x-1}+\frac{4}{x+2}=\frac{3}{x} $$

$$ \frac{x-5}{2}+\frac{2 x-1}{2+3 x}=\frac{5 x-1}{10}-\frac{7}{5} $$

$$ \frac{3-5 x}{x+2}=2+\frac{x-11}{x+4} $$

$$ \frac{3}{x+1}+\frac{7}{x+2}=\frac{6}{x-1} $$

$$ \frac{7}{x^{2}+x-12}-\frac{6}{x^{2}+2 x-8}=0 $$

$$ \frac{x^{2}-7 x+10}{x^{2}-7 x+12}=\frac{x^{2}+3 x-10}{x^{2}+3 x-8} $$

$$ \frac{x-3}{x^{2}-3 x-4}=\frac{x-1}{x^{2}-x-2} $$

$$ x^{2}+4 x-\frac{7}{x^{2}+4 x+5}=1 $$

$$ \frac{1}{x^{2}-3 x+3}+\frac{2}{x^{2}-3 x+4}=\frac{6}{x^{2}-3 x+5} $$

$$ \frac{1}{x-8}+\frac{1}{x-6}+\frac{1}{x+6}+\frac{1}{x+8}=0 $$

$$ \frac{2}{x-14}-\frac{5}{x-13}=\frac{2}{x-9}-\frac{5}{x-11} $$