Find the Roots (Zeros) C(t)=-0.005t^4+0.139t^3-0.051t^2+1.800t

$C (t) = - 0.005 t^{4} + 0.139 t^{3} - 0.051 t^{2} + 1.8 t$

Set $- 0.005 t^{4} + 0.139 t^{3} - 0.051 t^{2} + 1.8 t$ equal to $0$ .

$- 0.005 t^{4} + 0.139 t^{3} - 0.051 t^{2} + 1.8 t = 0$

Solve for $x$ .

Factor $0.001 t$ out of $- 0.005 t^{4} + 0.139 t^{3} - 0.051 t^{2} + 1.8 t$ .

Factor $0.001 t$ out of $- 0.005 t^{4}$ .

$0.001 t (- 5 t^{3}) + 0.139 t^{3} - 0.051 t^{2} + 1.8 t = 0$

Factor $0.001 t$ out of $0.139 t^{3}$ .

$0.001 t (- 5 t^{3}) + 0.001 t (139 t^{2}) - 0.051 t^{2} + 1.8 t = 0$

Factor $0.001 t$ out of $- 0.051 t^{2}$ .

$0.001 t (- 5 t^{3}) + 0.001 t (139 t^{2}) + 0.001 t (- 51 t) + 1.8 t = 0$

Factor $0.001 t$ out of $1.8 t$ .

$0.001 t (- 5 t^{3}) + 0.001 t (139 t^{2}) + 0.001 t (- 51 t) + 0.001 t (1800) = 0$

Factor $0.001 t$ out of $0.001 t (- 5 t^{3}) + 0.001 t (139 t^{2})$ .

$0.001 t (- 5 t^{3} + 139 t^{2}) + 0.001 t (- 51 t) + 0.001 t (1800) = 0$

Factor $0.001 t$ out of $0.001 t (- 5 t^{3} + 139 t^{2}) + 0.001 t (- 51 t)$ .

$0.001 t (- 5 t^{3} + 139 t^{2} - 51 t) + 0.001 t (1800) = 0$

Factor $0.001 t$ out of $0.001 t (- 5 t^{3} + 139 t^{2} - 51 t) + 0.001 t (1800)$ .

$0.001 t (- 5 t^{3} + 139 t^{2} - 51 t + 1800) = 0$

Divide each term in $0.001 t (- 5 t^{3} + 139 t^{2} - 51 t + 1800) = 0$ by $0.001$ and simplify.

Divide each term in $0.001 t (- 5 t^{3} + 139 t^{2} - 51 t + 1800) = 0$ by $0.001$ .

$\frac{0.001 t (- 5 t^{3} + 139 t^{2} - 51 t + 1800)}{0.001} = \frac{0}{0.001}$

Simplify the left side.

Cancel the common factor of $0.001$ .

Cancel the common factor.

$\frac{0.001 t (- 5 t^{3} + 139 t^{2} - 51 t + 1800)}{0.001} = \frac{0}{0.001}$

Divide $t (- 5 t^{3} + 139 t^{2} - 51 t + 1800)$ by $1$ .

$t (- 5 t^{3} + 139 t^{2} - 51 t + 1800) = \frac{0}{0.001}$

Apply the distributive property.

$t (- 5 t^{3}) + t (139 t^{2}) + t (- 51 t) + t \cdot 1800 = \frac{0}{0.001}$

Simplify.

Rewrite using the commutative property of multiplication.

$- 5 t \cdot t^{3} + t (139 t^{2}) + t (- 51 t) + t \cdot 1800 = \frac{0}{0.001}$

Rewrite using the commutative property of multiplication.

$- 5 t \cdot t^{3} + 139 t \cdot t^{2} + t (- 51 t) + t \cdot 1800 = \frac{0}{0.001}$

Rewrite using the commutative property of multiplication.

$- 5 t \cdot t^{3} + 139 t \cdot t^{2} - 51 t \cdot t + t \cdot 1800 = \frac{0}{0.001}$

Move $1800$ to the left of $t$ .

$- 5 t \cdot t^{3} + 139 t \cdot t^{2} - 51 t \cdot t + 1800 \cdot t = \frac{0}{0.001}$

Simplify each term.

Multiply $t$ by $t^{3}$ by adding the exponents.

Move $t^{3}$ .

$- 5 (t^{3} t) + 139 t \cdot t^{2} - 51 t \cdot t + 1800 \cdot t = \frac{0}{0.001}$

Multiply $t^{3}$ by $t$ .

Raise $t$ to the power of $1$ .

$- 5 (t^{3} t^{1}) + 139 t \cdot t^{2} - 51 t \cdot t + 1800 \cdot t = \frac{0}{0.001}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 5 t^{3 + 1} + 139 t \cdot t^{2} - 51 t \cdot t + 1800 \cdot t = \frac{0}{0.001}$

Add $3$ and $1$ .

$- 5 t^{4} + 139 t \cdot t^{2} - 51 t \cdot t + 1800 \cdot t = \frac{0}{0.001}$

Multiply $t$ by $t^{2}$ by adding the exponents.

Move $t^{2}$ .

$- 5 t^{4} + 139 (t^{2} t) - 51 t \cdot t + 1800 \cdot t = \frac{0}{0.001}$

Multiply $t^{2}$ by $t$ .

Raise $t$ to the power of $1$ .

$- 5 t^{4} + 139 (t^{2} t^{1}) - 51 t \cdot t + 1800 \cdot t = \frac{0}{0.001}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 5 t^{4} + 139 t^{2 + 1} - 51 t \cdot t + 1800 \cdot t = \frac{0}{0.001}$

Add $2$ and $1$ .

$- 5 t^{4} + 139 t^{3} - 51 t \cdot t + 1800 \cdot t = \frac{0}{0.001}$

Multiply $t$ by $t$ by adding the exponents.

Move $t$ .

$- 5 t^{4} + 139 t^{3} - 51 (t \cdot t) + 1800 \cdot t = \frac{0}{0.001}$

Multiply $t$ by $t$ .

$- 5 t^{4} + 139 t^{3} - 51 t^{2} + 1800 \cdot t = \frac{0}{0.001}$

$- 5 t^{4} + 139 t^{3} - 51 t^{2} + 1800 t = \frac{0}{0.001}$

Simplify the right side.

Divide $0$ by $0.001$ .

$- 5 t^{4} + 139 t^{3} - 51 t^{2} + 1800 t = 0$

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 0$

$q = \pm 1, \pm 5$

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$0$

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

$- 5 {(0)}^{4} + 139 {(0)}^{3} - 51 {(0)}^{2} + 1800 (0)$

Simplify the expression. In this case, the expression is equal to $0$ so $t = 0$ is a root of the polynomial.

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$- 5 \cdot 0 + 139 {(0)}^{3} - 51 {(0)}^{2} + 1800 (0)$

Multiply $- 5$ by $0$ .

$0 + 139 {(0)}^{3} - 51 {(0)}^{2} + 1800 (0)$

Raising $0$ to any positive power yields $0$ .

$0 + 139 \cdot 0 - 51 {(0)}^{2} + 1800 (0)$

Multiply $139$ by $0$ .

$0 + 0 - 51 {(0)}^{2} + 1800 (0)$

Raising $0$ to any positive power yields $0$ .

$0 + 0 - 51 \cdot 0 + 1800 (0)$

Multiply $- 51$ by $0$ .

$0 + 0 + 0 + 1800 (0)$

Multiply $1800$ by $0$ .

$0 + 0 + 0 + 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$0 + 0 + 0$

Add $0$ and $0$ .

$0 + 0$

Add $0$ and $0$ .

$0$

Since $0$ is a known root, divide the polynomial by $t + 0$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- 5 t^{4} + 139 t^{3} - 51 t^{2} + 1800 t}{t + 0}$

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$0$	$- 5$	$139$	$- 51$	$1800$	$0$
	\xa0				\xa0

The first number in the dividend $(- 5)$ is put into the first position of the result area (below the horizontal line).

$0$	$- 5$	$139$	$- 51$	$1800$	$0$
	\xa0				\xa0
	$- 5$

Multiply the newest entry in the result $(- 5)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(139)$ .

$0$	$- 5$	$139$	$- 51$	$1800$	$0$
	\xa0	$0$			\xa0
	$- 5$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$0$	$- 5$	$139$	$- 51$	$1800$	$0$
	\xa0	$0$			\xa0
	$- 5$	$139$

Multiply the newest entry in the result $(139)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(- 51)$ .

$0$	$- 5$	$139$	$- 51$	$1800$	$0$
	\xa0	$0$	$0$		\xa0
	$- 5$	$139$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$0$	$- 5$	$139$	$- 51$	$1800$	$0$
	\xa0	$0$	$0$		\xa0
	$- 5$	$139$	$- 51$

Multiply the newest entry in the result $(- 51)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(1800)$ .

$0$	$- 5$	$139$	$- 51$	$1800$	$0$
	\xa0	$0$	$0$	$0$	\xa0
	$- 5$	$139$	$- 51$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$0$	$- 5$	$139$	$- 51$	$1800$	$0$
	\xa0	$0$	$0$	$0$	\xa0
	$- 5$	$139$	$- 51$	$1800$

Multiply the newest entry in the result $(1800)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$0$	$- 5$	$139$	$- 51$	$1800$	$0$
	\xa0	$0$	$0$	$0$	$0$
	$- 5$	$139$	$- 51$	$1800$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$0$	$- 5$	$139$	$- 51$	$1800$	$0$
	\xa0	$0$	$0$	$0$	$0$
	$- 5$	$139$	$- 51$	$1800$	$0$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$- 5 t^{3} + 139 t^{2} + (- 51) t + 1800$

Simplify the quotient polynomial.

$- 5 t^{3} + 139 t^{2} - 51 t + 1800$

Graph each side of the equation. The solution is the x-value of the point of intersection.

$t \approx 27.89695095$

The polynomial can be written as a set of linear factors.

$(t + 0) (t - 27.89695095)$

These are the roots (zeros) of the polynomial $- 5 t^{4} + 139 t^{3} - 51 t^{2} + 1800 t$ .

$t = 0, 27.89695095$

Find the Roots (Zeros) C(t)=-0.005t^4+0.139t^3-0.051t^2+1.800t

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