Prove That$$5 x ^ { 4 } + 2 x ^ { 3 } - 1 \text { is } \Theta \left( x ^ { 4 } \right)$$

Question

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The Answer is

5 x ^ { 4 } + 2 x ^ { 3 } - 1

is

O \left( x ^ { 4 } \right)

since

\left| 5 x ^ { 4 } + 2 x ^ { 3 } - 1 \right| \leq \left| 5 x ^ { 4 } + 2 x ^ { 4 } \right| \leq 7 \left| x ^ { 4 } \right|

(if

\left. x \geq 1 \right)

. Also,

x ^ { 4 }

is

O \left( 5 x ^ { 4 } + 2 x ^ { 3 } - 1 \right.

) since

\left| x ^ { 4 } \right| \leq \left| 5 x ^ { 4 } + x ^ { 3 } \right| \leq \left| 5 x ^ { 4 } + 2 x ^ { 3 } - 1 \right|

(if

x \geq 1

).

Prove That $5 x ^ { 4 } + 2 x ^ { 3 } - 1 \text { is } \Theta \left( x ^ { 4 } \right)$