About The Fundamental Theorem of Algebra:
The Fundamental Theorem of Algebra along with the Complete Factorization Theorem tells us that a polynomial of degree n will have exactly 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 satisfies the given conditions.
$$a)\hspace{.2em}f(2)=-30$$ $$\text{Zeros}: 5, 4, -3$$
$$b)\hspace{.2em}f(1)=-24$$ $$\text{Zeros}: 9, -2, -1$$
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#2:
Instructions: Write a polynomial function of degree 3 that satisfies the given conditions.
$$a)\hspace{.2em}f(1)=-8$$ $$\text{Zeros}: 2 \hspace{.25em}\text{multiplicity}\hspace{.25em}2, 5$$
$$b)\hspace{.2em}f(-1)=-32$$ $$\text{Zeros}: 1\hspace{.25em}\text{multiplicity}\hspace{.25em}2, -5$$
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#3:
Instructions: Write a polynomial function of degree 3 that satisfies the given conditions.
$$a)\hspace{.2em}f(2)=135$$ $$\text{Zeros}: -1 \hspace{.25em}\text{multiplicity}\hspace{.25em}3$$
$$b)\hspace{.2em}f(0)=-252$$ $$\text{Zeros}: -6, 3, -2$$
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#4:
Instructions: Write a polynomial function of degree 3 that satisfies the given conditions.
$$a)\hspace{.2em}f(3)=20$$ $$\text{Zeros}: -7, -1, 2$$
$$b)\hspace{.2em}f(1)=1$$ $$\text{Zeros}: \frac{1}{2}, \frac{3}{4}, -1$$
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#5:
Instructions: Write a polynomial function of degree 3 that satisfies the given conditions.
$$a)\hspace{.2em}f(-1)=-16$$ $$\text{Zeros}: 3 \hspace{.25em}\text{multiplicity}\hspace{.25em}2, -\frac{1}{2}$$
$$b)\hspace{.2em}f(0)=-25$$ $$\text{Zeros}: -10 \hspace{.25em}\text{multiplicity}\hspace{.25em}2, \frac{1}{4}$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}f(x)=-x^3 + 6x^2 + 7x - 60$$
$$b)\hspace{.2em}f(x)=\frac{1}{2}x^3 - 3x^2 - \frac{25}{2}x - 9$$
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#2:
Solutions:
$$a)\hspace{.2em}f(x)=2x^3 - 18x^2 + 48x - 40$$
$$b)\hspace{.2em}f(x)=-2x^3 - 6x^2 + 18x - 10$$
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#3:
Solutions:
$$a)\hspace{.2em}f(x)=5x^3 + 15x^2 + 15x + 5$$
$$b)\hspace{.2em}f(x)=7x^3 + 35x^2 - 84x - 252$$
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#4:
Solutions:
$$a)\hspace{.2em}f(x)=\frac{1}{2}x^3 + 3x^2 - \frac{9}{2}x - 7$$
$$b)\hspace{.2em}f(x)=4x^3-x^2-\frac{7}{2}x + \frac{3}{2}$$
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#5:
Solutions:
$$a)\hspace{.2em}f(x)=2x^3 - 11x^2 + 12x + 9$$
$$b)\hspace{.2em}f(x)=x^3 + \frac{79}{4}x^2 + 95x - 25$$
