McGraw Hill Integrated II, 2012
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McGraw Hill Integrated II, 2012 View details
10. Roots and Zeros
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Exercise 14 Page 77

If a is a zero of f(x)=0, then (x-a) is a factor of f(x).

f(x)=x^4-5x^3+5x^2+5x-6

Practice makes perfect

We want to write a polynomial function with integral coefficients so that f(x)=0 has the given zeros. 3, -1, 1, 2 Recall that if a is a zero of f(x)=0, then (x-a) is a factor of f(x).

Root Factor
3 x-3
-1 x-(-1)
1 x-1
2 x-2
Polynomial f(x)= (x-3) [x-(-1)] (x-1)(x-2)

Let's simplify the polynomial by applying the Distributive Property. For simplicity, we will start by multiplying the first two factors and the last two factors separately. (x-3) & * [x-(-1)] (x-1) & * (x-2) After we find these products, we will multiply the obtained expressions.

(x-3)(x-(-1))
(x-3)(x+1)
x(x+1)-3(x+1)
â–¼
Distribute x & - 3
x^2+x-3(x+1)
x^2+x-3x-3
x^2-2x-3

Let's continue by finding the product of the last two factors.

(x-1)(x-2)
x(x-2)-1(x-2)
â–¼
Distribute x & - 1
x^2-2x-1(x-2)
x^2-2x-x+2
x^2-3x+2

Finding the product of these two polynomials will give us the desired polynomial function.

f(x)=(x^2-2x-3)(x^2-3x+2)
f(x)=x^2(x^2-3x+2)-2x(x^2-3x+2)-3(x^2-3x+2)
â–¼
Distribute x^2 & - 2x & -3
f(x)=x^4-3x^3+2x^2-2x(x^2-3x+2)-3(x^2-3x+2)
f(x)=x^4-3x^3+2x^2-2x^3+6x^2-4x-3(x^2-3x+2)
f(x)=x^4-3x^3+2x^2-2x^3+6x^2-4x-3x^2+9x-6
f(x)=x^4-5x^3+5x^2+5x-6