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A cubic polynomial function pp with integer coefficients has exactly two xx-intercepts at (−2,0)(-2, 0) and (1,0)(1, 0) in the xyxy-plane. The graph of y=p(x)y = p(x) is tangent to the xx-axis at one of these intercepts and intersects the yy-axis at (0,−6)(0, -6). What is the remainder when p(x)p(x) is divided by x+3x + 3?

  1. 48Cevap
  2. B
    -48
  3. C
    -6
  4. D
    -60

Cevap

48
The correct answer is 48. A cubic polynomial p(x)p(x) with exactly two xx-intercepts at (−2,0)(-2, 0) and (1,0)(1, 0) that is tangent to the xx-axis at one of these intercepts must have one factor of multiplicity 2. This gives two possible forms: p(x)=a(x+2)(x−1)2p(x) = a(x+2)(x-1)^2 or p(x)=b(x+2)2(x−1)p(x) = b(x+2)^2(x-1). Using the yy-intercept (0,−6)(0, -6), we can solve for the coefficients: if p(x)=a(x+2)(x−1)2p(x) = a(x+2)(x-1)^2, then p(0)=2a=−6p(0) = 2a = -6, which gives a=−3a = -3. If p(x)=b(x+2)2(x−1)p(x) = b(x+2)^2(x-1), then p(0)=−4b=−6p(0) = -4b = -6, which gives b=1.5b = 1.5. Since the polynomial must have integer coefficients, the correct function is p(x)=−3(x+2)(x−1)2p(x) = -3(x+2)(x-1)^2. By the Remainder Theorem, dividing p(x)p(x) by x+3x + 3 yields a remainder equal to p(−3)p(-3). Substituting -3 into the function gives p(−3)=−3(−3+2)(−3−1)2=−3(−1)(16)=48p(-3) = -3(-3+2)(-3-1)^2 = -3(-1)(16) = 48.

Adım Adım Çözüm

1
Identify the general form of the cubic polynomial based on its x-intercepts and their multiplicities.
The polynomial must be of the form p(x)=a(x+2)(x−1)2p(x) = a(x+2)(x-1)^2 or p(x)=b(x+2)2(x−1)p(x) = b(x+2)^2(x-1).
Since there are exactly two x-intercepts and the graph is tangent to the x-axis at one of them, one of the factors must have a multiplicity of 2.
2
Determine the coefficients using the y-intercept (0, -6) and check which function has integer coefficients.
Evaluating at x = 0 gives either 2a=−6⇒a=−32a = -6 \Rightarrow a = -3 or −4b=−6⇒b=1.5-4b = -6 \Rightarrow b = 1.5. Thus, the correct polynomial with integer coefficients is p(x)=−3(x+2)(x−1)2p(x) = -3(x+2)(x-1)^2.
The y-intercept gives the value of the function at x = 0. The constraint requires integer coefficients, which rules out b = 1.5.
3
Apply the Remainder Theorem to find the remainder when p(x) is divided by x + 3.
The remainder is p(−3)=−3(−3+2)(−3−1)2=−3(−1)(−4)2=48p(-3) = -3(-3+2)(-3-1)^2 = -3(-1)(-4)^2 = 48.
According to the Remainder Theorem, the remainder of a polynomial p(x) divided by x - c is p(c). Here, c = -3.

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