Answer:
D) f(x) = x³ +2x² +5x -26
Step-by-step explanation:
You want the polynomial function of least degree with integer coefficients that has roots 2, -2+3i.
Roots
A polynomial function with real coefficients will only have complex roots in conjugate pairs. This means the function has another root that is -2-3i.
The sum of the three roots is ...
2 + (-2+3i) + (-2 -3i) = -2
The coefficient of the x² term is the opposite of this: +2. (Eliminates choices A and C.)
The product of the roots is ...
(2)(-2 +3i)(-2 -3i) = 2((-2)² -(3i)²) = 2(4 +9) = 26
The constant in the polynomial is the opposite of this, -26. (Eliminates choice B.)
Now, we know the correct answer choice must be choice D.
Factors
Each root p corresponds to a factor (x -p). This means the factored form of the polynomial can be written as ...
f(x) = (x -2)(x -(-2+3i))(x -(-2-3i))
We can rearrange the factors to ...
f(x) = (x -2)((x +2) -3i)((x +2) +3i) . . . . . product of factors
f(x) = (x -2)((x +2)² -(3i)²) . . . . . . . right pair simplified
f(x) = (x -2)(x² +4x +4 +9) = x(x² +4x +13) -2(x² +4x +13)
f(x) = x³ +4x² +13x -2x² -8x -26 . . . . . . eliminate parentheses
f(x) = x³ +2x² +5x -26 . . . . . . . . . . . collect terms; matches choice D)
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Additional comment
In expanding the factored form, we have made use of the factoring of the difference of squares:
a² -b² = (a -b)(a +b)
Of course, the distributive property is used to eliminate parentheses.
