Simplify:
(i) $2x^2 (x^3 - x) - 3x (x^4 + 2x) -2(x^4 - 3x^2)$
(ii) $x^3y (x^2 - 2x) + 2xy (x^3 - x^4)$
(iii) $3a^2 + 2 (a + 2) - 3a (2a + 1)$
(iv) $x (x + 4) + 3x (2x^2 - 1) + 4x^2 + 4$
(v) $a (b-c) - b (c - a) - c (a - b)$
(vi) $a (b - c) + b (c - a) + c (a - b)$
(vii) $4ab (a - b) - 6a^2 (b - b^2) -3b^2 (2a^2 - a) + 2ab (b-a)$
(viii) $x^2 (x^2 + 1) - x^3 (x + 1) - x (x^3 - x)$
(ix) $2a^2 + 3a (1 - 2a^3) + a (a + 1)$
(x) $a^2 (2a - 1) + 3a + a^3 - 8$
(xi) $\frac{3}{2}-x^2 (x^2 - 1) + \frac{1}{4}-x^2 (x^2 + x) - \frac{3}{4}x (x^3 - 1)$
(xii) $a^2b (a - b^2) + ab^2 (4ab - 2a^2) - a^3b (1 - 2b)$
(xiii) $a^2b (a^3 - a + 1) - ab (a^4 - 2a^2 + 2a) - b (a^3- a^2 -1)$.
To do:
We have to simplify the given expressions.
Solution:
(i) $2x^2 (x^3 -x) - 3x (x^4 + 2x) -2 (x^4 - 3x^2) = 2x^2(x^3)-2x^2(x)-3x(x^4)-3x( 2x)-2x^4 + 6x^2$
$= 2x^{2+3}-2x^{2+1}-3x^{1+4}-6x^{1+1}-2x^4 + 6x^2$
$= 2x^5 - 2x^3 - 3x^5 - 6x^2 - 2x^4 + 6x^2$
$= 2x^5 - 3x^5 - 2x^4 - 2x^3 + 6x^2 - 6x^2$
$= -x^5 - 2x^4 - 2x^3$
(ii) $x^{3} y(x^{2}-2 x)+2 x y(x^{3}-x^{4})=x^{5} y-2 x^{4} y+2 x^{4} y-2 x^{5} y$
$=x^{5} y-2 x^{5} y-2 x^{4} y+2 x^{4} y$
$=-x^{5} y$
(iii) $3a^2 + 2 (a + 2) - 3a (2a + 1)=3a^2 + 2 (a) + 2(2) - 3a (2a) -3a(1)$
$=3a^2 + 2a + 4 - 6a^2 - 3a$
$= 3a^2 - 6a^2 + 2a - 3a + 4$
$= -3a^2 - a + 4$
(iv) $x(x+4)+3 x(2 x^{2}-1)+4 x^{2}+4=x^{2}+4 x+6 x^{3}-3 x+4 x^{2}+4$
$=x^{2}+4 x^{2}+4 x-3 x+6 x^{3}+4$
$=5 x^{2}+x+6 x^{3}+4$
(v) $a (b-c) - b (c - a) - c (a - b)=ab - ac - bc+ ab - ac + bc$
$= 2ab - 2ac$
(vi) $a (b - c) + b (c - a) + c (a - b)=ab - ac + bc - ab + ac - bc$
$= ab - ab + bc - bc + ac - ac$
$= 0$
(vii) $4ab (a - b) - 6a^2 (b - b^2) -3b^2 (2a^2 - a) + 2ab (b-a)=4a^2b - 4ab^2 - 6a^2b + 6a^2b^2 - 6a^2b^2 + 3ab^2 + 2ab^2 - 2a^2b$
$= 4a^2b - 6a^2b - 2a^2b - 4ab^2 + 3ab^2 + 2ab^2 + 6a^2b^2 - 6a^2b^2$
$= 4a^2b - 8a^2b - 4ab^2 + 5 ab^2 + 0$
$=- 4a^2b + ab^2$
(viii) $x^2 (x^2 + 1) - x^3 (x + 1) - x (x^3 - x)$
$= x^4 + x^2-x^4-x^3-x^4 + x^2$
$= x^4-x^4-x^4-x^3 + x^2 + x^2$
$= -x^4- x^3 + 2x^2$
(ix) $2a^2 + 3a (1 - 2a^3) + a (a + 1)=2a^2 + 3 a - 3 a (2a^3) + a^2 + a$
$= 2a^2 + 3a - 6a^{1+3} + a^2 + a$
$= 2a^2 + 3a - 6a^4 + a^2 + a$
$= -6a^4 + 3a^2 + 4a$
(x) $a^2 (2a - 1) + 3a + a^3 - 8=a^2( 2a) - a^2(1)+3a + a^3-8$
$= 2a^3 - a^2 + 3a + a^3 - 8$
$= 2a^3 + a^3 - a^2 + 3a - 8$
$= 3a^3 - a^2 + 3a - 8$
(xi) $\frac{3}{2} x^{2}(x^{2}-1)+\frac{1}{4} x^{2}(x^{2}+x)-\frac{3}{4} x(x^{3}-1)$
$=\frac{3}{2} x^{2+2}-\frac{3}{2} x^{2}+\frac{1}{4} x^{2+2}+\frac{1}{4} x^{2+1}-\frac{3}{4} x^{1+3}+\frac{3}{4} x$
$=\frac{3}{2} x^{4}-\frac{3}{2} x^{2}+\frac{1}{4} x^{4}+\frac{1}{4} x^{3}-\frac{3}{4} x^{4}+\frac{3}{4} x$
$=\frac{3}{2} x^{4}+\frac{1}{4} x^{4}-\frac{3}{4} x^{4}+\frac{1}{4} x^{3}-\frac{3}{2} x^{2}+\frac{3}{4} x$
$=\frac{6 x^{4}+x^{4}-3 x^{4}}{4}+\frac{1}{4} x^{3}-\frac{3}{2} x^{2}+\frac{3}{4} x$
$=\frac{4 x^{4}}{4}+\frac{1}{4} x^{3}-\frac{3}{2} x^{2}+\frac{3}{4} x$
$=x^{4}+\frac{1}{4} x^{3}-\frac{3}{2} x^{2}+\frac{3}{4} x$
(xii) $a^2b (a - b^2) + ab^2 (4ab - 2a^2) - a^3b (1 - 2b)$
$= a^{2+1}b-a^2b^{2+1}+ 4a^{1 +1} b^{2 +1}-2a^{2+1}b^2-a^3b + 2a^3b^{1+1}$
$= a^3b - a^2b^3 + 4a^2b^3 - 2a^3b^2 - a^3b + 2a^3b^2$
$= a^3b - a^3b - a^2b^3 + 4a^2b^3 - 2a^3b^2 + 2a^3b^2$
$= 0 + 3a^2b^3 + 0$
$= 3a^2b^3$
(xiii) $a^{2} b(a^{3}-a+1)-a b(a^{4}-2 a^{2}+2 a)-b(a^{3}-a^{2}-1)$
$=a^{5} b-a^{5} b-a^{3} b+2 a^{3} b-a^{3} b+a^{2} b-2 a^{2} b+a^{2} b+b $
$=b$
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