Evaluate
\frac{\left(1-2x\right)\left(3x-4\right)\left(x-2\right)^{3}}{2}
Expand
-3x^{5}+\frac{47x^{4}}{2}-71x^{3}+102x^{2}-68x+16
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\left(3x-4\right)\left(-x+\frac{1}{2}\right)\left(x^{3}-6x^{2}+12x-8\right)
Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-2\right)^{3}.
\left(3x\left(-x\right)+\frac{3}{2}x-4\left(-x\right)-2\right)\left(x^{3}-6x^{2}+12x-8\right)
Use the distributive property to multiply 3x-4 by -x+\frac{1}{2}.
\left(3x\left(-x\right)+\frac{3}{2}x+4x-2\right)\left(x^{3}-6x^{2}+12x-8\right)
Multiply -4 and -1 to get 4.
\left(3x\left(-x\right)+\frac{11}{2}x-2\right)\left(x^{3}-6x^{2}+12x-8\right)
Combine \frac{3}{2}x and 4x to get \frac{11}{2}x.
3\left(-x\right)x^{4}-18x^{3}\left(-x\right)+36x^{2}\left(-x\right)-24x\left(-x\right)+\frac{11}{2}x^{4}-35x^{3}+78x^{2}-68x+16
Use the distributive property to multiply 3x\left(-x\right)+\frac{11}{2}x-2 by x^{3}-6x^{2}+12x-8 and combine like terms.
3\left(-x\right)x^{4}+18x^{3}x+36x^{2}\left(-x\right)-24x\left(-x\right)+\frac{11}{2}x^{4}-35x^{3}+78x^{2}-68x+16
Multiply -18 and -1 to get 18.
3\left(-x\right)x^{4}+18x^{4}+36x^{2}\left(-x\right)-24x\left(-x\right)+\frac{11}{2}x^{4}-35x^{3}+78x^{2}-68x+16
To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.
3\left(-x\right)x^{4}+18x^{4}+36x^{2}\left(-x\right)+24xx+\frac{11}{2}x^{4}-35x^{3}+78x^{2}-68x+16
Multiply -24 and -1 to get 24.
3\left(-x\right)x^{4}+18x^{4}+36x^{2}\left(-x\right)+24x^{2}+\frac{11}{2}x^{4}-35x^{3}+78x^{2}-68x+16
Multiply x and x to get x^{2}.
3\left(-x\right)x^{4}+\frac{47}{2}x^{4}+36x^{2}\left(-x\right)+24x^{2}-35x^{3}+78x^{2}-68x+16
Combine 18x^{4} and \frac{11}{2}x^{4} to get \frac{47}{2}x^{4}.
3\left(-x\right)x^{4}+\frac{47}{2}x^{4}+36x^{2}\left(-x\right)+102x^{2}-35x^{3}-68x+16
Combine 24x^{2} and 78x^{2} to get 102x^{2}.
-3xx^{4}+\frac{47}{2}x^{4}+36x^{2}\left(-1\right)x+102x^{2}-35x^{3}-68x+16
Multiply 3 and -1 to get -3.
-3x^{5}+\frac{47}{2}x^{4}+36x^{2}\left(-1\right)x+102x^{2}-35x^{3}-68x+16
To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.
-3x^{5}+\frac{47}{2}x^{4}+36x^{3}\left(-1\right)+102x^{2}-35x^{3}-68x+16
To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.
-3x^{5}+\frac{47}{2}x^{4}-36x^{3}+102x^{2}-35x^{3}-68x+16
Multiply 36 and -1 to get -36.
-3x^{5}+\frac{47}{2}x^{4}-71x^{3}+102x^{2}-68x+16
Combine -36x^{3} and -35x^{3} to get -71x^{3}.
\left(3x-4\right)\left(-x+\frac{1}{2}\right)\left(x^{3}-6x^{2}+12x-8\right)
Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-2\right)^{3}.
\left(3x\left(-x\right)+\frac{3}{2}x-4\left(-x\right)-2\right)\left(x^{3}-6x^{2}+12x-8\right)
Use the distributive property to multiply 3x-4 by -x+\frac{1}{2}.
\left(3x\left(-x\right)+\frac{3}{2}x+4x-2\right)\left(x^{3}-6x^{2}+12x-8\right)
Multiply -4 and -1 to get 4.
\left(3x\left(-x\right)+\frac{11}{2}x-2\right)\left(x^{3}-6x^{2}+12x-8\right)
Combine \frac{3}{2}x and 4x to get \frac{11}{2}x.
3\left(-x\right)x^{4}-18x^{3}\left(-x\right)+36x^{2}\left(-x\right)-24x\left(-x\right)+\frac{11}{2}x^{4}-35x^{3}+78x^{2}-68x+16
Use the distributive property to multiply 3x\left(-x\right)+\frac{11}{2}x-2 by x^{3}-6x^{2}+12x-8 and combine like terms.
3\left(-x\right)x^{4}+18x^{3}x+36x^{2}\left(-x\right)-24x\left(-x\right)+\frac{11}{2}x^{4}-35x^{3}+78x^{2}-68x+16
Multiply -18 and -1 to get 18.
3\left(-x\right)x^{4}+18x^{4}+36x^{2}\left(-x\right)-24x\left(-x\right)+\frac{11}{2}x^{4}-35x^{3}+78x^{2}-68x+16
To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.
3\left(-x\right)x^{4}+18x^{4}+36x^{2}\left(-x\right)+24xx+\frac{11}{2}x^{4}-35x^{3}+78x^{2}-68x+16
Multiply -24 and -1 to get 24.
3\left(-x\right)x^{4}+18x^{4}+36x^{2}\left(-x\right)+24x^{2}+\frac{11}{2}x^{4}-35x^{3}+78x^{2}-68x+16
Multiply x and x to get x^{2}.
3\left(-x\right)x^{4}+\frac{47}{2}x^{4}+36x^{2}\left(-x\right)+24x^{2}-35x^{3}+78x^{2}-68x+16
Combine 18x^{4} and \frac{11}{2}x^{4} to get \frac{47}{2}x^{4}.
3\left(-x\right)x^{4}+\frac{47}{2}x^{4}+36x^{2}\left(-x\right)+102x^{2}-35x^{3}-68x+16
Combine 24x^{2} and 78x^{2} to get 102x^{2}.
-3xx^{4}+\frac{47}{2}x^{4}+36x^{2}\left(-1\right)x+102x^{2}-35x^{3}-68x+16
Multiply 3 and -1 to get -3.
-3x^{5}+\frac{47}{2}x^{4}+36x^{2}\left(-1\right)x+102x^{2}-35x^{3}-68x+16
To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.
-3x^{5}+\frac{47}{2}x^{4}+36x^{3}\left(-1\right)+102x^{2}-35x^{3}-68x+16
To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.
-3x^{5}+\frac{47}{2}x^{4}-36x^{3}+102x^{2}-35x^{3}-68x+16
Multiply 36 and -1 to get -36.
-3x^{5}+\frac{47}{2}x^{4}-71x^{3}+102x^{2}-68x+16
Combine -36x^{3} and -35x^{3} to get -71x^{3}.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}