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gates of the arctic national park and preserve - GetMeFoodie

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 5**Question:** A palynologist is analyzing pollen data and models the growth of a particular type of pollen count using the polynomial \( P(x) = 3x^3 - 5x + 4 \). Determine the sum of the roots of \( P(x) \).

📌 Solution:** To find the sum of the roots of the polynomial \( P(x) = 3x^3 - 5x + 4 \), we use Vieta's formulas. For a cubic polynomial of the form \( ax^3 + bx^2 + cx + d \), the sum of the roots is given by \( -\frac{b}{a} \). Here, \( a = 3 \), \( b = 0 \), \( c = -5 \), and \( d = 4 \). Therefore, the sum of the roots is:

📌 -\frac{b}{a} = -\frac{0}{3} = 0

📌 Thus, the sum of the roots is \(\boxed{0}\).

📌 Question:** A mathematician working on algebraic topology defines a function \( f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 \). Determine the remainder when \( f(x) \) is divided by \( x - 1 \).

📌 Solution:** To find the remainder when \( f(x) \) is divided by \( x - 1 \), we use the Remainder Theorem, which states that the remainder of the division of a polynomial \( f(x) \) by \( x - c \) is \( f(c) \). Here, \( c = 1 \), so we calculate \( f(1) \):

📌 f(1) = 1^4 - 4 \cdot 1^3 + 6 \cdot 1^2 - 4 \cdot 1 + 1 = 1 - 4 + 6 - 4 + 1 = 0

📌 The remainder is \(\boxed{0}\).

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