Asked by john
3x(x- 7)(x-1)^ 2
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GPT-5 mini
AI
Expand: 3x(x - 7)(x - 1)^2
1) (x - 1)^2 = x^2 - 2x + 1
2) (x - 7)(x^2 - 2x + 1) = x^3 - 9x^2 + 15x - 7
3) Multiply by 3x: 3x(x^3 - 9x^2 + 15x - 7) = 3x^4 - 27x^3 + 45x^2 - 21x
So the expanded form is:
3x^4 - 27x^3 + 45x^2 - 21x
Additional info:
- Factored form (given): 3x(x - 7)(x - 1)^2
- Roots and multiplicities: x = 0 (mult. 1), x = 7 (mult. 1), x = 1 (mult. 2)
- Degree: 4; leading coefficient: 3 (>0) → both ends of the graph go to +∞
- x-intercepts: (0,0), (7,0), (1,0) with the graph touching and turning at x = 1 (because multiplicity 2) and crossing at x = 0 and x = 7.
1) (x - 1)^2 = x^2 - 2x + 1
2) (x - 7)(x^2 - 2x + 1) = x^3 - 9x^2 + 15x - 7
3) Multiply by 3x: 3x(x^3 - 9x^2 + 15x - 7) = 3x^4 - 27x^3 + 45x^2 - 21x
So the expanded form is:
3x^4 - 27x^3 + 45x^2 - 21x
Additional info:
- Factored form (given): 3x(x - 7)(x - 1)^2
- Roots and multiplicities: x = 0 (mult. 1), x = 7 (mult. 1), x = 1 (mult. 2)
- Degree: 4; leading coefficient: 3 (>0) → both ends of the graph go to +∞
- x-intercepts: (0,0), (7,0), (1,0) with the graph touching and turning at x = 1 (because multiplicity 2) and crossing at x = 0 and x = 7.
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