Part F. Use the rule or identity you created in part E to find the factors for the expressions in the table below.

| Expression | Factors |
|---|---|
| [tex]x^2+100[/tex] | |
| [tex]x^2+36[/tex] | |
| [tex]16 x^2+9[/tex] | |

The expressions are factored using the identity for sums of squares. The factors are: x 2 + 100 = ( x + 10 i ) ( x − 10 i ) , x 2 + 36 = ( x + 6 i ) ( x − 6 i ) , and 16 x 2 + 9 = ( 4 x + 3 i ) ( 4 x − 3 i ) .
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Answered by Anonymous | 2025-07-04

Use the identity a 2 + b 2 = ( a + bi ) ( a − bi ) to factor the expressions.
For x 2 + 100 , the factors are ( x + 10 i ) and ( x − 10 i ) .
For x 2 + 36 , the factors are ( x + 6 i ) and ( x − 6 i ) .
For 16 x 2 + 9 , the factors are ( 4 x + 3 i ) and ( 4 x − 3 i ) .

The factors are:
x 2 + 100 = ( x + 10 i ) ( x − 10 i )
x 2 + 36 = ( x + 6 i ) ( x − 6 i )
16 x 2 + 9 = ( 4 x + 3 i ) ( 4 x − 3 i )
Explanation

Understanding the Problem and the Identity We are asked to factor the given expressions using the rule or identity from part E. Let's assume the identity from part E is based on the difference of squares, extended to handle sums of squares using complex numbers. Specifically, we'll use the identity a 2 + b 2 = ( a + bi ) ( a − bi ) , where i is the imaginary unit, i 2 = − 1 .

Factoring x 2 + 100 For the expression x 2 + 100 , we can identify a = x and b = 10 . Applying the identity, we get: x 2 + 100 = ( x + 10 i ) ( x − 10 i ) So, the factors are ( x + 10 i ) and ( x − 10 i ) .

Factoring x 2 + 36 For the expression x 2 + 36 , we can identify a = x and b = 6 . Applying the identity, we get: x 2 + 36 = ( x + 6 i ) ( x − 6 i ) So, the factors are ( x + 6 i ) and ( x − 6 i ) .

Factoring 16 x 2 + 9 For the expression 16 x 2 + 9 , we can identify a = 4 x and b = 3 . Applying the identity, we get: 16 x 2 + 9 = ( 4 x + 3 i ) ( 4 x − 3 i ) So, the factors are ( 4 x + 3 i ) and ( 4 x − 3 i ) .

Final Factors Therefore, the factors for the given expressions are:



x 2 + 100 = ( x + 10 i ) ( x − 10 i )
x 2 + 36 = ( x + 6 i ) ( x − 6 i )
16 x 2 + 9 = ( 4 x + 3 i ) ( 4 x − 3 i )

Examples
Complex numbers might seem abstract, but they're incredibly useful in fields like electrical engineering and quantum mechanics. For example, when analyzing alternating current (AC) circuits, complex numbers help represent the impedance, which combines resistance and reactance. Factoring expressions involving sums of squares using complex numbers, as we did here, allows engineers to simplify circuit analysis and design more efficient systems. This same math is crucial in quantum mechanics for describing wave functions and understanding the behavior of particles at the atomic level.

Answered by GinnyAnswer | 2025-07-04