Concept explainers
To write: A polynomial function f of least degree that has rational coefficients, a leading coefficient of 1 and the given zeros.
The required polynomial function f would be f(x)=x3−6x2+5x+12 .
Given information:
Zeros of a function: −1, 3, 4 .
Formula used:
If x=a, b, c are zeros of the function f(x) , then (x−a), (x−b) and (x−c) would be factors of the function.
Calculation:
For the given problem, there are 3 zeros. This means there will be 3 factors of the given function as:
f(x)=(x−(−1))(x−3)(x−4)=(x+1)(x−3)(x−4)
Now expand the factors using the distributive property (a+b)(c+d)=a⋅c+a⋅d+b⋅c+b⋅d:
f(x)=(x+1)(x−3)(x−4)=(x(x−3)+1(x−3))(x−4)=(x2−3x+x−3)(x−4)=(x2−2x−3)(x−4)=x(x2−2x−3)−4(x2−2x−3)=x3−2x2−3x−4x2+8x+12=x3−6x2+5x+12
Therefore, the required function would be f(x)=x3−6x2+5x+12 .
Chapter 2 Solutions
Holt Mcdougal Larson Algebra 2: Student Edition 2012
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