Solve y=380t(1-t)^3-270t^2(1-t)^2+260t^3(1-t)

$$y = 380 t ( 1 - t ) ^ { 3 } - 270 t ^ { 2 } ( 1 - t ) ^ { 2 } + 260 t ^ { 3 } ( 1 - t )$$

Solve for y

$$y=10t\left(1-t\right)\left(91t^{2}-103t+38\right)$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(1-t\right)^{3}.

$$y=380t\left(1-3t+3t^{2}-t^{3}\right)-270t^{2}\left(1-t\right)^{2}+260t^{3}\left(1-t\right)$$

Use the distributive property to multiply 380t by 1-3t+3t^{2}-t^{3}.

$$y=380t-1140t^{2}+1140t^{3}-380t^{4}-270t^{2}\left(1-t\right)^{2}+260t^{3}\left(1-t\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-t\right)^{2}.

$$y=380t-1140t^{2}+1140t^{3}-380t^{4}-270t^{2}\left(1-2t+t^{2}\right)+260t^{3}\left(1-t\right)$$

Use the distributive property to multiply 260t^{3} by 1-t.

$$y=380t-1140t^{2}+1140t^{3}-380t^{4}-270t^{2}\left(1-2t+t^{2}\right)+260t^{3}-260t^{4}$$

Use the distributive property to multiply -270t^{2} by 1-2t+t^{2}.

$$y=380t-1140t^{2}+1140t^{3}-380t^{4}-270t^{2}+540t^{3}-270t^{4}+260t^{3}-260t^{4}$$

Combine -1140t^{2} and -270t^{2} to get -1410t^{2}.

$$y=380t-1410t^{2}+1140t^{3}-380t^{4}+540t^{3}-270t^{4}+260t^{3}-260t^{4}$$

Combine 1140t^{3} and 540t^{3} to get 1680t^{3}.

$$y=380t-1410t^{2}+1680t^{3}-380t^{4}-270t^{4}+260t^{3}-260t^{4}$$

Combine -380t^{4} and -270t^{4} to get -650t^{4}.

$$y=380t-1410t^{2}+1680t^{3}-650t^{4}+260t^{3}-260t^{4}$$

Combine 1680t^{3} and 260t^{3} to get 1940t^{3}.

$$y=380t-1410t^{2}+1940t^{3}-650t^{4}-260t^{4}$$

Combine -650t^{4} and -260t^{4} to get -910t^{4}.

$$y=380t-1410t^{2}+1940t^{3}-910t^{4}$$