Exercise 17 Page 494

By the Factor Theorem the expression x- a is a factor of a polynomial if and only if the value a is a zero of the related polynomial function. In other words, the expression x- a is a factor of a polynomial if and only if the graph of the related polynomial function has an x-intercept at ( a,0). We are given three points. ( - 3,0), (- 1,0), (1,0)We know that the graph of a polynomial has x-intercepts at these points. Therefore, the expressions x-( - 3), x-(- 1), and x-1 must be the factors of the polynomial. Let's multiply these three binomials.

(x-( - 3))(x-(- 1))(x-1)

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Multiply

SubNeg

a-(- b)=a+b

(x+3)(x+1)(x-1)

ExpandDiffSquares

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

(x+3)(x^2-1^2)

BaseOne

1^a=1

(x+3)(x^2-1)

Distr

Distribute (x^2-1)

x(x^2-1)+3(x^2-1)

Distr

Distribute x & 3

x^3-x+3x^2-3

CommutativePropAdd

Commutative Property of Addition

x^3+3x^2-x-3

The degree of a polynomial is the highest degree of its monomial. x^3+3x^2-x-3 The degree of the above polynomial is 3. We obtained this polynomial by multiplying binomials that must be the factors of the described polynomial, so the expression x^3+3x^2-x-3 also must be the factor of that polynomial. Thus, the least possible degree of the described polynomial is 3.