Exercise 43 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 -2 and 1+sqrt(7).Because 1+sqrt(7) is a zero, by the Irrational Conjugates Theorem 1-sqrt(7) is also a zero. This means we have three zeros, so the least degree is 3. Let's write the factored form of f. f(x)= 1(x-( -2))(x-( 1+sqrt(7)))(x-(1-sqrt(7))) Finally, we will simplify the equation by multiplying the factors.
   f(x)=1(x-(-2))(x-(1+sqrt(7)))(x-(1-sqrt(7)))

   IdPropMult

   Identity Property of Multiplication

   f(x)=(x-(-2))(x-(1+sqrt(7)))(x-(1-sqrt(7)))

   Distr

   Distribute - 1

   f(x)=(x+2)(x-1-sqrt(7))(x-1+sqrt(7))

   AddPar

   Add parentheses

   f(x)=(x+2)((x-1)-sqrt(7))((x-1)+sqrt(7))

   ExpandDiffSquares

   (a+b)(a-b)=a^2-b^2

   f(x)=(x+2)((x-1)^2-(sqrt(7))^2)

   PowSqrt

   ( sqrt(a) )^2 = a

   f(x)=(x+2)((x-1)^2-7)

   ExpandNegPerfectSquare

   (a-b)^2=a^2-2ab+b^2

   f(x)=(x+2)(x^2-2x+1-7)

   SubTerm

   Subtract term

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

   ▼

   Multiply

   Distr

   Distribute (x^2-2x-6)

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

   Distr

   Distribute x

   f(x)=x^3-2x^2-6x+2(x^2-2x-6)

   Distr

   Distribute 2

   f(x)=x^3-2x^2-6x+2x^2-4x-12

   AddSubTerms

   Add and subtract terms

   f(x)=x^3-10x-12