Find the Roots (Zeros) f(t)=t^2(3t^2-10t+7)

$f (t) = t^{2} (3 t^{2} - 10 t + 7)$

Set $t^{2} (3 t^{2} - 10 t + 7)$ equal to $0$ .

$t^{2} (3 t^{2} - 10 t + 7) = 0$

Solve for $x$ .

Simplify $t^{2} (3 t^{2} - 10 t + 7)$ .

Apply the distributive property.

$t^{2} (3 t^{2}) + t^{2} (- 10 t) + t^{2} \cdot 7 = 0$

Simplify.

Rewrite using the commutative property of multiplication.

$3 t^{2} t^{2} + t^{2} (- 10 t) + t^{2} \cdot 7 = 0$

Rewrite using the commutative property of multiplication.

$3 t^{2} t^{2} - 10 t^{2} t + t^{2} \cdot 7 = 0$

Move $7$ to the left of $t^{2}$ .

$3 t^{2} t^{2} - 10 t^{2} t + 7 \cdot t^{2} = 0$

Simplify each term.

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

Move $t^{2}$ .

$3 (t^{2} t^{2}) - 10 t^{2} t + 7 \cdot t^{2} = 0$

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

$3 t^{2 + 2} - 10 t^{2} t + 7 \cdot t^{2} = 0$

Add $2$ and $2$ .

$3 t^{4} - 10 t^{2} t + 7 \cdot t^{2} = 0$

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

Move $t$ .

$3 t^{4} - 10 (t \cdot t^{2}) + 7 \cdot t^{2} = 0$

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

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

$3 t^{4} - 10 (t^{1} t^{2}) + 7 \cdot t^{2} = 0$

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

$3 t^{4} - 10 t^{1 + 2} + 7 \cdot t^{2} = 0$

Add $1$ and $2$ .

$3 t^{4} - 10 t^{3} + 7 \cdot t^{2} = 0$

$3 t^{4} - 10 t^{3} + 7 t^{2} = 0$

Factor the left side of the equation.

Factor $t^{2}$ out of $3 t^{4} - 10 t^{3} + 7 t^{2}$ .

Factor $t^{2}$ out of $3 t^{4}$ .

$t^{2} (3 t^{2}) - 10 t^{3} + 7 t^{2} = 0$

Factor $t^{2}$ out of $- 10 t^{3}$ .

$t^{2} (3 t^{2}) + t^{2} (- 10 t) + 7 t^{2} = 0$

Factor $t^{2}$ out of $7 t^{2}$ .

$t^{2} (3 t^{2}) + t^{2} (- 10 t) + t^{2} \cdot 7 = 0$

Factor $t^{2}$ out of $t^{2} (3 t^{2}) + t^{2} (- 10 t)$ .

$t^{2} (3 t^{2} - 10 t) + t^{2} \cdot 7 = 0$

Factor $t^{2}$ out of $t^{2} (3 t^{2} - 10 t) + t^{2} \cdot 7$ .

$t^{2} (3 t^{2} - 10 t + 7) = 0$

Factor.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 7 = 21$ and whose sum is $b = - 10$ .

Factor $- 10$ out of $- 10 t$ .

$t^{2} (3 t^{2} - 10 t + 7) = 0$

Rewrite $- 10$ as $- 3$ plus $- 7$

$t^{2} (3 t^{2} + (- 3 - 7) t + 7) = 0$

Apply the distributive property.

$t^{2} (3 t^{2} - 3 t - 7 t + 7) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$t^{2} ((3 t^{2} - 3 t) - 7 t + 7) = 0$

Factor out the greatest common factor (GCF) from each group.

$t^{2} (3 t (t - 1) - 7 (t - 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $t - 1$ .

$t^{2} ((t - 1) (3 t - 7)) = 0$

Remove unnecessary parentheses.

$t^{2} (t - 1) (3 t - 7) = 0$

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

$t^{2} = 0$

$t - 1 = 0$

$3 t - 7 = 0$

Set $t^{2}$ equal to $0$ and solve for $t$ .

Set $t^{2}$ equal to $0$ .

$t^{2} = 0$

Solve $t^{2} = 0$ for $t$ .

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$t = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$t = \pm \sqrt{0^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$t = \pm 0$

Plus or minus $0$ is $0$ .

$t = 0$

Set $t - 1$ equal to $0$ and solve for $t$ .

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

$t - 1 = 0$

Add $1$ to both sides of the equation.

$t = 1$

Set $3 t - 7$ equal to $0$ and solve for $t$ .

Set $3 t - 7$ equal to $0$ .

$3 t - 7 = 0$

Solve $3 t - 7 = 0$ for $t$ .

Add $7$ to both sides of the equation.

$3 t = 7$

Divide each term in $3 t = 7$ by $3$ and simplify.

Divide each term in $3 t = 7$ by $3$ .

$\frac{3 t}{3} = \frac{7}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 t}{3} = \frac{7}{3}$

Divide $t$ by $1$ .

$t = \frac{7}{3}$

The final solution is all the values that make $t^{2} (t - 1) (3 t - 7) = 0$ true.

$t = 0, 1, \frac{7}{3}$

Find the Roots (Zeros) f(t)=t^2(3t^2-10t+7)

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