Find the Roots (Zeros) h(t)=(t-8)^1(t-4)^2(t-2)^3(t-1)^4

$h (t) = {(t - 8)}^{1} {(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4}$

Set ${(t - 8)}^{1} {(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4}$ equal to $0$ .

${(t - 8)}^{1} {(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4} = 0$

Solve for $x$ .

Simplify ${(t - 8)}^{1} {(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4}$ .

Simplify.

$(t - 8) {(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4} = 0$

Apply the distributive property.

$(t {(t - 4)}^{2} - 8 {(t - 4)}^{2}) {(t - 2)}^{3} {(t - 1)}^{4} = 0$

Apply the distributive property.

$(t {(t - 4)}^{2} {(t - 2)}^{3} - 8 {(t - 4)}^{2} {(t - 2)}^{3}) {(t - 1)}^{4} = 0$

Apply the distributive property.

$t {(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4} - 8 {(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4} = 0$

Factor ${(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4}$ out of $t {(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4} - 8 {(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4}$ .

Factor ${(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4}$ out of $t {(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4}$ .

${(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4} (t) - 8 {(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4} = 0$

Factor ${(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4}$ out of $- 8 {(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4}$ .

${(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4} (t) + {(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4} (- 8) = 0$

Factor ${(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4}$ out of ${(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4} (t) + {(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4} (- 8)$ .

${(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4} (t - 8) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(t - 4)}^{2} = 0$

${(t - 2)}^{3} = 0$

${(t - 1)}^{4} = 0$

$t - 8 = 0$

Set ${(t - 4)}^{2}$ equal to $0$ and solve for $t$ .

Set ${(t - 4)}^{2}$ equal to $0$ .

${(t - 4)}^{2} = 0$

Solve ${(t - 4)}^{2} = 0$ for $t$ .

Set the $t - 4$ equal to $0$ .

$t - 4 = 0$

Add $4$ to both sides of the equation.

$t = 4$

Set ${(t - 2)}^{3}$ equal to $0$ and solve for $t$ .

Set ${(t - 2)}^{3}$ equal to $0$ .

${(t - 2)}^{3} = 0$

Solve ${(t - 2)}^{3} = 0$ for $t$ .

Set the $t - 2$ equal to $0$ .

$t - 2 = 0$

Add $2$ to both sides of the equation.

$t = 2$

Set ${(t - 1)}^{4}$ equal to $0$ and solve for $t$ .

Set ${(t - 1)}^{4}$ equal to $0$ .

${(t - 1)}^{4} = 0$

Solve ${(t - 1)}^{4} = 0$ for $t$ .

Set the $t - 1$ equal to $0$ .

$t - 1 = 0$

Add $1$ to both sides of the equation.

$t = 1$

Set $t - 8$ equal to $0$ and solve for $t$ .

Set $t - 8$ equal to $0$ .

$t - 8 = 0$

Add $8$ to both sides of the equation.

$t = 8$

The final solution is all the values that make ${(t - 4)}^{2} {(t - 2)}^{3} {(t - 1)}^{4} (t - 8) = 0$ true.

$t = 4, 2, 1, 8$

Find the Roots (Zeros) h(t)=(t-8)^1(t-4)^2(t-2)^3(t-1)^4

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