Question

Construct a polynomial function with the following properties: fifth degree, 2 is a zero of multiplicity 4, −2 is the only other zero, leading coefficient is 3.

Answer 1

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Answer:

  p(x) = 3(x -2)^4(x +2) . . . in factored form

  p(x) = 3x^5 -18x^4 +24x^3 +48x^2 -144x +96  . . . in standard form

Step-by-step explanation:

When a polynomial has a zero at x=p, it has a factor of (x -p). The exponent of the factor is the multiplicity of the zero. The product of factors can be multiplied by 3 to make the leading coefficient be 3.

  p(x) = 3(x -2)^4(x -(-2))

  = 3(x^4 +4(-2)x^3 +6(-2)^2x^2 +4(-2)^3x +(-2)^4)·(x +2)

  = 3(x^4 -8x^3 +24x^2 -32x +16)(x +2)

  = 3(x^5 -6x^4 +8x^3 +16x^2 -48x +32)

  p(x) = 3x^5 -18x^4 +24x^3 +48x^2 -144x +96