Question

Let p be a polynomial function defined by p(x) = 2x^3+5x^2-6x-9. Use the fact that p(-4)=-33, p(-3)=0, p(-2)=7, and p(-1)=0 to rewrite the expression for p(x) as the product of linear factors.

Answer 1

Answer:

  p(x) = (2x -3)(x+3)(x +1)

Step-by-step explanation:

p(-3) = 0 tells you that (x+3) is a factor.

p(-1) = 0 tells you that (x+1) is a factor.

The remaining factor can be (ax +b), so the function is ...

  p(x) = (ax +b)(x +3)(x +1)

Matching leading coefficients, we have ...

  ax³ = 2x³   ⇒   a = 2

Matching constant terms, we have ...

  b(3)(1) = -9    ⇒   b = -3

So, our factorization is ...

  p(x) = (2x -3)(x +3)(x +1)

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Check

We can see if our factored form matches the other two data points given.

  p(-4) = (-11)(-1)(-3) = -33 . . . . tells us the vertical scaling is correct

  p(-2) = (-7)(1)(-1) = 7 . . . . . . . additional check passes

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The graph shows the original function (red) and the factored function (blue dots). They are identical.