What is expand (1+x+x^2)^4 ?

The solution to expand (1+x+x^2)^4 is x^8+4x^7+10x^6+16x^5+19x^4+16x^3+10x^2+4x+1

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expand (1+x+x^2)^4

expand

(1 + x + x^{2})^{4}

x^{8} + 4 x^{7} + 10 x^{6} + 16 x^{5} + 19 x^{4} + 16 x^{3} + 10 x^{2} + 4 x + 1

Solution steps

(1 + x + x^{2})^{4}

Simplify

(1 + x + x^{2})^{4} : (1 + x + x^{2}) (1 + x + x^{2}) (1 + x + x^{2}) (1 + x + x^{2})

= (1 + x + x^{2}) (1 + x + x^{2}) (1 + x + x^{2}) (1 + x + x^{2})

Expand

(1 + x + x^{2}) (1 + x + x^{2}) : x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1

= (x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1) (1 + x + x^{2}) (1 + x + x^{2})

Expand

(x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1) (1 + x + x^{2}) : x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 1

= (x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 1) (1 + x + x^{2})

Distribute parentheses

= x^{6} \cdot 1 + x^{6} x + x^{6} x^{2} + 3 x^{5} \cdot 1 + 3 x^{5} x + 3 x^{5} x^{2} + 6 x^{4} \cdot 1 + 6 x^{4} x + 6 x^{4} x^{2} + 7 x^{3} \cdot 1 + 7 x^{3} x + 7 x^{3} x^{2} + 6 x^{2} \cdot 1 + 6 x^{2} x + 6 x^{2} x^{2} + 3 x \cdot 1 + 3 xx + 3 x x^{2} + 1 \cdot 1 + 1 \cdot x + 1 \cdot x^{2}

Simplify

x^{6} \cdot 1 + x^{6} x + x^{6} x^{2} + 3 x^{5} \cdot 1 + 3 x^{5} x + 3 x^{5} x^{2} + 6 x^{4} \cdot 1 + 6 x^{4} x + 6 x^{4} x^{2} + 7 x^{3} \cdot 1 + 7 x^{3} x + 7 x^{3} x^{2} + 6 x^{2} \cdot 1 + 6 x^{2} x + 6 x^{2} x^{2} + 3 x \cdot 1 + 3 xx + 3 x x^{2} + 1 \cdot 1 + 1 \cdot x + 1 \cdot x^{2} : x^{8} + 4 x^{7} + 10 x^{6} + 16 x^{5} + 19 x^{4} + 16 x^{3} + 10 x^{2} + 4 x + 1

= x^{8} + 4 x^{7} + 10 x^{6} + 16 x^{5} + 19 x^{4} + 16 x^{3} + 10 x^{2} + 4 x + 1

2 \times \frac{2}{2}

32 \times 8

- 3 \times \frac{3}{2}

2 \cdot + 18

x^{2} - 22 x - 48

What is expand (1+x+x^2)^4 ?
The solution to expand (1+x+x^2)^4 is x^8+4x^7+10x^6+16x^5+19x^4+16x^3+10x^2+4x+1