What is expand (1+z+(z^2)/2)^4 ?

The solution to expand (1+z+(z^2)/2)^4 is (17z^4)/2+5z^5+2z^6+10z^3+8z^2+4z+1+(z^7)/2+(z^8)/(16)

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expand (1+z+(z^2)/2)^4

expand

(1 + z + \frac{z ^{2}}{2})^{4}

\frac{17 z ^{4}}{2} + 5 z^{5} + 2 z^{6} + 10 z^{3} + 8 z^{2} + 4 z + 1 + \frac{z ^{7}}{2} + \frac{z ^{8}}{16}

Solution steps

(1 + z + \frac{z ^{2}}{2})^{4}

Simplify

(1 + z + \frac{z ^{2}}{2})^{4} : (1 + z + \frac{z ^{2}}{2}) (1 + z + \frac{z ^{2}}{2}) (1 + z + \frac{z ^{2}}{2}) (1 + z + \frac{z ^{2}}{2})

= (1 + z + \frac{z ^{2}}{2}) (1 + z + \frac{z ^{2}}{2}) (1 + z + \frac{z ^{2}}{2}) (1 + z + \frac{z ^{2}}{2})

Expand

(1 + z + \frac{z ^{2}}{2}) (1 + z + \frac{z ^{2}}{2}) : 1 + 2 z + 2 z^{2} + z^{3} + \frac{z ^{4}}{4}

= (1 + 2 z + 2 z^{2} + z^{3} + \frac{z ^{4}}{4}) (1 + z + \frac{z ^{2}}{2}) (1 + z + \frac{z ^{2}}{2})

Expand

(1 + 2 z + 2 z^{2} + z^{3} + \frac{z ^{4}}{4}) (1 + z + \frac{z ^{2}}{2}) : \frac{9 z ^{4}}{4} + 4 z^{3} + \frac{9 z ^{2}}{2} + 3 z + 1 + \frac{3 z ^{5}}{4} + \frac{z ^{6}}{8}

= (\frac{9 z ^{4}}{4} + 4 z^{3} + \frac{9 z ^{2}}{2} + 3 z + 1 + \frac{3 z ^{5}}{4} + \frac{z ^{6}}{8}) (1 + z + \frac{z ^{2}}{2})

Distribute parentheses

= \frac{9 z ^{4}}{4} \cdot 1 + \frac{9 z ^{4}}{4} z + \frac{9 z ^{4}}{4} \cdot \frac{z ^{2}}{2} + 4 z^{3} \cdot 1 + 4 z^{3} z + 4 z^{3} \frac{z ^{2}}{2} + \frac{9 z ^{2}}{2} \cdot 1 + \frac{9 z ^{2}}{2} z + \frac{9 z ^{2}}{2} \cdot \frac{z ^{2}}{2} + 3 z \cdot 1 + 3 zz + 3 z \frac{z ^{2}}{2} + 1 \cdot 1 + 1 \cdot z + 1 \cdot \frac{z ^{2}}{2} + \frac{3 z ^{5}}{4} \cdot 1 + \frac{3 z ^{5}}{4} z + \frac{3 z ^{5}}{4} \cdot \frac{z ^{2}}{2} + \frac{z ^{6}}{8} \cdot 1 + \frac{z ^{6}}{8} z + \frac{z ^{6}}{8} \cdot \frac{z ^{2}}{2}

Simplify

\frac{9 z ^{4}}{4} \cdot 1 + \frac{9 z ^{4}}{4} z + \frac{9 z ^{4}}{4} \cdot \frac{z ^{2}}{2} + 4 z^{3} \cdot 1 + 4 z^{3} z + 4 z^{3} \frac{z ^{2}}{2} + \frac{9 z ^{2}}{2} \cdot 1 + \frac{9 z ^{2}}{2} z + \frac{9 z ^{2}}{2} \cdot \frac{z ^{2}}{2} + 3 z \cdot 1 + 3 zz + 3 z \frac{z ^{2}}{2} + 1 \cdot 1 + 1 \cdot z + 1 \cdot \frac{z ^{2}}{2} + \frac{3 z ^{5}}{4} \cdot 1 + \frac{3 z ^{5}}{4} z + \frac{3 z ^{5}}{4} \cdot \frac{z ^{2}}{2} + \frac{z ^{6}}{8} \cdot 1 + \frac{z ^{6}}{8} z + \frac{z ^{6}}{8} \cdot \frac{z ^{2}}{2} : \frac{17 z ^{4}}{2} + 5 z^{5} + 2 z^{6} + 10 z^{3} + 8 z^{2} + 4 z + 1 + \frac{z ^{7}}{2} + \frac{z ^{8}}{16}

= \frac{17 z ^{4}}{2} + 5 z^{5} + 2 z^{6} + 10 z^{3} + 8 z^{2} + 4 z + 1 + \frac{z ^{7}}{2} + \frac{z ^{8}}{16}

(16 u^{3} + v^{3})^{2} + 144 u^{8} v^{4} + 1

(x - e) (x + y - e)

(x) (x + 2 + 2)

\frac{( x - x ^{2} ) ( 1 + x )}{1 - x}

(- 9, 7) y (- 9, - 1)

What is expand (1+z+(z^2)/2)^4 ?
The solution to expand (1+z+(z^2)/2)^4 is (17z^4)/2+5z^5+2z^6+10z^3+8z^2+4z+1+(z^7)/2+(z^8)/(16)