About The Fundamental Theorem of Algebra:
The fundamental theorem of algebra tells us that a polynomial of degree n, will have n complex solutions, although some of these solutions may be repeated.
Test Objectives
- Demonstrate the ability to write a polynomial function given certain conditions
#1:
Instructions: Write a polynomial function of degree 3 that satisifies the given conditions.
$$a)\hspace{.2em}f(2)=-30$$ $$\text{Zeros}: 5, 4, -3$$
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#2:
Instructions: Write a polynomial function of degree 3 that satisifies the given conditions.
$$a)\hspace{.2em}f(1)=-8$$ $$\text{Zeros}: 2 \hspace{.25em}\text{multiplicity}\hspace{.25em}2, 5$$
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#3:
Instructions: Write a polynomial function of degree 3 that satisifies the given conditions.
$$a)\hspace{.2em}f(2)=135$$ $$\text{Zeros}: -1 \hspace{.25em}\text{multiplicity}\hspace{.25em}3$$
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#4:
Instructions: Write a polynomial function of degree 3 that satisifies the given conditions.
$$a)\hspace{.2em}f(3)=20$$ $$\text{Zeros}: -7, -1, 2$$
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#5:
Instructions: Write a polynomial function of degree 3 that satisifies the given conditions.
$$a)\hspace{.2em}f(-1)=-16$$ $$\text{Zeros}: 3 \hspace{.25em}\text{multiplicity}\hspace{.25em}2, -\frac{1}{2}$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}f(x)=-x^3 + 6x^2 + 7x - 60$$
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#2:
Solutions:
$$a)\hspace{.2em}f(x)=2x^3 - 18x^2 + 48x - 40$$
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#3:
Solutions:
$$a)\hspace{.2em}f(x)=5x^3 + 15x^2 + 15x + 5$$
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#4:
Solutions:
$$a)\hspace{.2em}f(x)=\frac{1}{2}x^3 + 3x^2 - \frac{9}{2}x - 7$$
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#5:
Solutions:
$$a)\hspace{.2em}f(x)=2x^3 - 11x^2 + 12x + 9$$
