Exercise 42 Page 195

We want to write a polynomial function f that meets the following requirements.

1. It is of the least possible degree.
2. It has rational coefficients.
3. The leading coefficient is 1.
4. The zeros are - 4, - 2, and 5.If p is a zero of f, then (x-p) is a factor of f. With this, we can use the given zeros to write the factored form of f. Note that, because we have been given three zeros, the least degree is 3. Let's write the factored form of f. f(x)= 1(x-( -4))(x-( -2))(x-5) Finally, we will simplify the equation by multiplying the factors.
   f(x)=1(x-(-4))(x-(-2))(x-5)

   IdPropMult

   Identity Property of Multiplication

   f(x)=(x-(-4))(x-(-2))(x-5)

   NegNeg

   - (- a)=a

   f(x)=(x+4)(x+2)(x-5)

   MultII

   Multiply (x+4) by (x+2)

   f(x)=(x^2+2x+4x+8)(x-5)

   AddSubTerms

   Add and subtract terms

   f(x)=(x^2+6x+8)(x-5)

   MultII

   Multiply (x^2+6x+8) by (x-5)

   f(x)=x^3-5x^2+6x^2-30x+8x-40

   AddSubTerms

   Add and subtract terms

   f(x)=x^3+x^2-22x-40