Exercise 44 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 and 6-sqrt(7).Because 6-sqrt(7) is a zero, by the Irrational Conjugates Theorem 6+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- 4)(x-( 6-sqrt(7)))(x-(6+sqrt(7))) Finally, we will simplify the equation by multiplying the factors.
   f(x)=1(x-4)(x-(6-sqrt(7)))(x-(6+sqrt(7)))

   IdPropMult

   Identity Property of Multiplication

   f(x)=(x-4)(x-(6-sqrt(7)))(x-(6+sqrt(7)))

   Distr

   Distribute - 1

   f(x)=(x-4)(x-6+sqrt(7))(x-6-sqrt(7))

   AddPar

   Add parentheses

   f(x)=(x-4)((x-6)+sqrt(7))((x-6)-sqrt(7))

   ExpandDiffSquares

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

   f(x)=(x-4)((x-6)^2-(sqrt(7))^2)

   PowSqrt

   ( sqrt(a) )^2 = a

   f(x)=(x-4)((x-6)^2-7)

   ExpandNegPerfectSquare

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

   f(x)=(x-4)(x^2-12x+(6)^2-7)

   CalcPow

   Calculate power

   f(x)=(x-4)(x^2-12x+36-7)

   SubTerm

   Subtract term

   f(x)=(x-4)(x^2-12x+29)

   ▼

   Multiply

   Distr

   Distribute (x^2-12x+29)

   f(x)=x(x^2-12x+29)-4(x^2-12x+29)

   Distr

   Distribute x

   f(x)=x^3-12x^2+29x-4(x^2-12x+29)

   Distr

   Distribute -4

   f(x)=x^3-12x^2+29x-4x^2+48x-116

   AddSubTerms

   Add and subtract terms

   f(x)=x^3-16x^2+77x-116