Factor U(t)=2t^4+3t^3+48t^2+75t-50

$U (t) = 2 t^{4} + 3 t^{3} + 48 t^{2} + 75 t - 50$

Step 1

Regroup terms.

$U (t) = 3 t^{3} + 75 t + 2 t^{4} + 48 t^{2} - 50$

Step 2

Factor $3 t$ out of $3 t^{3} + 75 t$ .

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

$U (t) = 3 t (t^{2}) + 75 t + 2 t^{4} + 48 t^{2} - 50$

Factor $3 t$ out of $75 t$ .

$U (t) = 3 t (t^{2}) + 3 t (25) + 2 t^{4} + 48 t^{2} - 50$

Factor $3 t$ out of $3 t (t^{2}) + 3 t (25)$ .

$U (t) = 3 t (t^{2} + 25) + 2 t^{4} + 48 t^{2} - 50$

Step 3

Factor $2$ out of $2 t^{4} + 48 t^{2} - 50$ .

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

$U (t) = 3 t (t^{2} + 25) + 2 (t^{4}) + 48 t^{2} - 50$

Factor $2$ out of $48 t^{2}$ .

$U (t) = 3 t (t^{2} + 25) + 2 (t^{4}) + 2 (24 t^{2}) - 50$

Factor $2$ out of $- 50$ .

$U (t) = 3 t (t^{2} + 25) + 2 t^{4} + 2 (24 t^{2}) + 2 \cdot - 25$

Factor $2$ out of $2 t^{4} + 2 (24 t^{2})$ .

$U (t) = 3 t (t^{2} + 25) + 2 (t^{4} + 24 t^{2}) + 2 \cdot - 25$

Factor $2$ out of $2 (t^{4} + 24 t^{2}) + 2 \cdot - 25$ .

$U (t) = 3 t (t^{2} + 25) + 2 (t^{4} + 24 t^{2} - 25)$

Step 4

Rewrite $t^{4}$ as ${(t^{2})}^{2}$ .

$U (t) = 3 t (t^{2} + 25) + 2 ({(t^{2})}^{2} + 24 t^{2} - 25)$

Step 5

Let $u = t^{2}$ . Substitute $u$ for all occurrences of $t^{2}$ .

$U (t) = 3 t (t^{2} + 25) + 2 (u^{2} + 24 u - 25)$

Step 6

Factor $u^{2} + 24 u - 25$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 25$ and whose sum is $24$ .

$- 1, 25$

Write the factored form using these integers.

$U (t) = 3 t (t^{2} + 25) + 2 ((u - 1) (u + 25))$

Step 7

Replace all occurrences of $u$ with $t^{2}$ .

$U (t) = 3 t (t^{2} + 25) + 2 ((t^{2} - 1) (t^{2} + 25))$

Step 8

Rewrite $1$ as $1^{2}$ .

$U (t) = 3 t (t^{2} + 25) + 2 ((t^{2} - 1^{2}) (t^{2} + 25))$

Step 9

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = t$ and $b = 1$ .

$U (t) = 3 t (t^{2} + 25) + 2 ((t + 1) (t - 1) (t^{2} + 25))$

Remove unnecessary parentheses.

$U (t) = 3 t (t^{2} + 25) + 2 (t + 1) (t - 1) (t^{2} + 25)$

Step 10

Factor $t^{2} + 25$ out of $3 t (t^{2} + 25) + 2 (t + 1) (t - 1) (t^{2} + 25)$ .

Factor $t^{2} + 25$ out of $3 t (t^{2} + 25)$ .

$U (t) = (t^{2} + 25) (3 t) + 2 (t + 1) (t - 1) (t^{2} + 25)$

Factor $t^{2} + 25$ out of $2 (t + 1) (t - 1) (t^{2} + 25)$ .

$U (t) = (t^{2} + 25) (3 t) + (t^{2} + 25) (2 (t + 1) (t - 1))$

Factor $t^{2} + 25$ out of $(t^{2} + 25) (3 t) + (t^{2} + 25) (2 (t + 1) (t - 1))$ .

$U (t) = (t^{2} + 25) (3 t + 2 (t + 1) (t - 1))$

Step 11

Apply the distributive property.

$U (t) = (t^{2} + 25) (3 t + (2 t + 2 \cdot 1) (t - 1))$

Step 12

Multiply $2$ by $1$ .

$U (t) = (t^{2} + 25) (3 t + (2 t + 2) (t - 1))$

Step 13

Expand $(2 t + 2) (t - 1)$ using the FOIL Method.

Apply the distributive property.

$U (t) = (t^{2} + 25) (3 t + 2 t (t - 1) + 2 (t - 1))$

Apply the distributive property.

$U (t) = (t^{2} + 25) (3 t + 2 t \cdot t + 2 t \cdot - 1 + 2 (t - 1))$

Apply the distributive property.

$U (t) = (t^{2} + 25) (3 t + 2 t \cdot t + 2 t \cdot - 1 + 2 t + 2 \cdot - 1)$

Step 14

Simplify and combine like terms.

Simplify each term.

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

Move $t$ .

$U (t) = (t^{2} + 25) (3 t + 2 (t \cdot t) + 2 t \cdot - 1 + 2 t + 2 \cdot - 1)$

Multiply $t$ by $t$ .

$U (t) = (t^{2} + 25) (3 t + 2 t^{2} + 2 t \cdot - 1 + 2 t + 2 \cdot - 1)$

Multiply $- 1$ by $2$ .

$U (t) = (t^{2} + 25) (3 t + 2 t^{2} - 2 t + 2 t + 2 \cdot - 1)$

Multiply $2$ by $- 1$ .

$U (t) = (t^{2} + 25) (3 t + 2 t^{2} - 2 t + 2 t - 2)$

Add $- 2 t$ and $2 t$ .

$U (t) = (t^{2} + 25) (3 t + 2 t^{2} + 0 - 2)$

Add $2 t^{2}$ and $0$ .

$U (t) = (t^{2} + 25) (3 t + 2 t^{2} - 2)$

Step 15

Reorder terms.

$U (t) = (t^{2} + 25) (2 t^{2} + 3 t - 2)$

Step 16

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 $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = 3$ .

Factor $3$ out of $3 t$ .

$U (t) = (t^{2} + 25) (2 t^{2} + 3 (t) - 2)$

Rewrite $3$ as $- 1$ plus $4$

$U (t) = (t^{2} + 25) (2 t^{2} + (- 1 + 4) t - 2)$

Apply the distributive property.

$U (t) = (t^{2} + 25) (2 t^{2} - 1 t + 4 t - 2)$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$U (t) = (t^{2} + 25) ((2 t^{2} - 1 t) + 4 t - 2)$

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

$U (t) = (t^{2} + 25) (t (2 t - 1) + 2 (2 t - 1))$

Factor the polynomial by factoring out the greatest common factor, $2 t - 1$ .

$U (t) = (t^{2} + 25) ((2 t - 1) (t + 2))$

Remove unnecessary parentheses.

$U (t) = (t^{2} + 25) (2 t - 1) (t + 2)$

Factor U(t)=2t^4+3t^3+48t^2+75t-50

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