Question Factor the trigonometric expression $2 \cot^2 Y + 9 \cot Y + 10$ completely, where $2 \cot^2 Y + 9 \cot Y + 10 = (2 \cot Y + \square)(\cot Y + \square)$ .

Studdy Solution

STEP 1

Assumptions1. The given expression is a quadratic trinomial in terms of $\cot Y$ . . The quadratic trinomial can be factored into two binomial expressions.

STEP 2

We need to factor the quadratic trinomial. The general form of a factored quadratic trinomial is $(ax + b)(cx + d)$ where $x$ is the variable, in this case $\cot Y$ .

STEP 3

We need to find values for $a$ , $b$ , $c$ , and $d$ such that1. $a \cdot c =2$ , the coefficient of $\cot^2 Y$ .
2. $b \cdot d =10$ , the constant term.
3. $a \cdot d + b \cdot c =9$ , the coefficient of $\cot Y$ .

STEP 4

We can start by finding pairs of factors for $2$ and $10$ .
For $2$ , the pairs of factors are $(1,2)$ and $(-1, -2)$ .
For $10$ , the pairs of factors are $(1,10)$ , $(-1, -10)$ , $(2,)$ , and $(-2, -)$ .

STEP 5

Now, we need to find a combination of these pairs that satisfies the third condition, $a \cdot d + b \cdot c =9$ .
After trying different combinations, we find that the pair $(2,1)$ for $a$ and $c$ , and the pair $(5,2)$ for $b$ and $d$ satisfies the condition.

STEP 6

Now, we can write the factored form of the given expression.
$2 \cot ^{2} Y+9 \cot Y+10 = (2 \cot Y +5)(\cot Y +2)$ This is the factored form of the given expression.

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Question Factor the trigonometric expression $2 \cot^2 Y + 9 \cot Y + 10$ completely, where $2 \cot^2 Y + 9 \cot Y + 10 = (2 \cot Y + \square)(\cot Y + \square)$ .

Studdy Solution

STEP 1

Assumptions1. The given expression is a quadratic trinomial in terms of $\cot Y$ . . The quadratic trinomial can be factored into two binomial expressions.

STEP 2

We need to factor the quadratic trinomial. The general form of a factored quadratic trinomial is $(ax + b)(cx + d)$ where $x$ is the variable, in this case $\cot Y$ .

STEP 3

We need to find values for $a$ , $b$ , $c$ , and $d$ such that1. $a \cdot c =2$ , the coefficient of $\cot^2 Y$ .
2. $b \cdot d =10$ , the constant term.
3. $a \cdot d + b \cdot c =9$ , the coefficient of $\cot Y$ .

STEP 4

We can start by finding pairs of factors for $2$ and $10$ .
For $2$ , the pairs of factors are $(1,2)$ and $(-1, -2)$ .
For $10$ , the pairs of factors are $(1,10)$ , $(-1, -10)$ , $(2,)$ , and $(-2, -)$ .

STEP 5

Now, we need to find a combination of these pairs that satisfies the third condition, $a \cdot d + b \cdot c =9$ .
After trying different combinations, we find that the pair $(2,1)$ for $a$ and $c$ , and the pair $(5,2)$ for $b$ and $d$ satisfies the condition.

STEP 6

Now, we can write the factored form of the given expression.
$2 \cot ^{2} Y+9 \cot Y+10 = (2 \cot Y +5)(\cot Y +2)$ This is the factored form of the given expression.

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Question Factor the trigonometric expression 2cot⁡2Y+9cot⁡Y+102 \cot^2 Y + 9 \cot Y + 102cot2Y+9cotY+10 completely, where 2cot⁡2Y+9cot⁡Y+10=(2cot⁡Y+□)(cot⁡Y+□)2 \cot^2 Y + 9 \cot Y + 10 = (2 \cot Y + \square)(\cot Y + \square)2cot2Y+9cotY+10=(2cotY+□)(cotY+□).

Studdy Solution

STEP 1

Assumptions1. The given expression is a quadratic trinomial in terms of cot⁡Y\cot YcotY. . The quadratic trinomial can be factored into two binomial expressions.

STEP 2

We need to factor the quadratic trinomial. The general form of a factored quadratic trinomial is(ax+b)(cx+d)(ax + b)(cx + d)(ax+b)(cx+d)where xxx is the variable, in this case cot⁡Y\cot YcotY.

STEP 3

We need to find values for aaa, bbb, ccc, and ddd such that1. a⋅c=2a \cdot c =2a⋅c=2, the coefficient of cot⁡2Y\cot^2 Ycot2Y.2. b⋅d=10b \cdot d =10b⋅d=10, the constant term.3. a⋅d+b⋅c=9a \cdot d + b \cdot c =9a⋅d+b⋅c=9, the coefficient of cot⁡Y\cot YcotY.

STEP 4

We can start by finding pairs of factors for 222 and 101010.For 222, the pairs of factors are (1,2)(1,2)(1,2) and (−1,−2)(-1, -2)(−1,−2).For 101010, the pairs of factors are (1,10)(1,10)(1,10), (−1,−10)(-1, -10)(−1,−10), (2,)(2,)(2,), and (−2,−)(-2, -)(−2,−).

STEP 5

Now, we need to find a combination of these pairs that satisfies the third condition, a⋅d+b⋅c=9a \cdot d + b \cdot c =9a⋅d+b⋅c=9.After trying different combinations, we find that the pair (2,1)(2,1)(2,1) for aaa and ccc, and the pair (5,2)(5,2)(5,2) for bbb and ddd satisfies the condition.

STEP 6

Now, we can write the factored form of the given expression.2cot⁡2Y+9cot⁡Y+10=(2cot⁡Y+5)(cot⁡Y+2)2 \cot ^{2} Y+9 \cot Y+10 = (2 \cot Y +5)(\cot Y +2)2cot2Y+9cotY+10=(2cotY+5)(cotY+2)This is the factored form of the given expression.

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Question Factor the trigonometric expression $2 \cot^2 Y + 9 \cot Y + 10$ completely, where $2 \cot^2 Y + 9 \cot Y + 10 = (2 \cot Y + \square)(\cot Y + \square)$ .

Assumptions1. The given expression is a quadratic trinomial in terms of $\cot Y$ . . The quadratic trinomial can be factored into two binomial expressions.

We need to factor the quadratic trinomial. The general form of a factored quadratic trinomial is $(ax + b)(cx + d)$ where $x$ is the variable, in this case $\cot Y$ .

We need to find values for $a$ , $b$ , $c$ , and $d$ such that1. $a \cdot c =2$ , the coefficient of $\cot^2 Y$ .
2. $b \cdot d =10$ , the constant term.
3. $a \cdot d + b \cdot c =9$ , the coefficient of $\cot Y$ .

We can start by finding pairs of factors for $2$ and $10$ .
For $2$ , the pairs of factors are $(1,2)$ and $(-1, -2)$ .
For $10$ , the pairs of factors are $(1,10)$ , $(-1, -10)$ , $(2,)$ , and $(-2, -)$ .

Now, we need to find a combination of these pairs that satisfies the third condition, $a \cdot d + b \cdot c =9$ .
After trying different combinations, we find that the pair $(2,1)$ for $a$ and $c$ , and the pair $(5,2)$ for $b$ and $d$ satisfies the condition.

Now, we can write the factored form of the given expression.
$2 \cot ^{2} Y+9 \cot Y+10 = (2 \cot Y +5)(\cot Y +2)$ This is the factored form of the given expression.