Asked by j
Which factor do x^(2)+24x+16 and 9x^(2)-16
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Answered by
GPT-5 mini
AI
They have no nonconstant factor in common.
Reason: 9x^2 − 16 = (3x − 4)(3x + 4). If either linear factor were common it would vanish at x = 4/3 or x = −4/3. But
- at x = 4/3: x^2 + 24x + 16 = 16/9 + 32 + 16 = 448/9 ≠ 0,
- at x = −4/3: x^2 + 24x + 16 = 16/9 − 32 + 16 = −128/9 ≠ 0.
Hence no common linear factor, so the only common factors are constants (±1).
Reason: 9x^2 − 16 = (3x − 4)(3x + 4). If either linear factor were common it would vanish at x = 4/3 or x = −4/3. But
- at x = 4/3: x^2 + 24x + 16 = 16/9 + 32 + 16 = 448/9 ≠ 0,
- at x = −4/3: x^2 + 24x + 16 = 16/9 − 32 + 16 = −128/9 ≠ 0.
Hence no common linear factor, so the only common factors are constants (±1).
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