1.
Factor completely the following expression: \(P(x)=x^4-8x^3+5x^2+74x-120\).
Test rational zeros. \(P(2)=0\), so \((x-2)\) is a factor. After factoring, test \(x=5\), so \((x-5)\) is also a factor. The remaining quadratic factors as \(x^2-x-12=(x-4)(x+3)\). Therefore, \(P(x)=(x-2)(x+3)(x-4)(x-5)\). Answer: B
2.
Solve the following polynomial equation by factoring: \(x^3-4x^2-7x+10=0\).
Test \(x=1\): \(1-4-7+10=0\), so \((x-1)\) is a factor. Dividing gives \(x^2-3x-10\), which factors as \((x-5)(x+2)\). Thus \((x-1)(x-5)(x+2)=0\), so \(x=1,5,-2\). Answer: A
3.
Write a polynomial equation with roots of \(\frac12\), \(-\sqrt{3}\), and \(\sqrt{3}\).
The root \(x=\frac12\) gives factor \(2x-1\). The roots \(x=-\sqrt{3}\) and \(x=\sqrt{3}\) give factors \((x+\sqrt{3})(x-\sqrt{3})=x^2-3\). Therefore, \(y=(2x-1)(x^2-3)\). Answer: C
4.
Determine the degree of the polynomial equation \(y=2(x-3)^2(x+4)^3\).
The degree is the sum of the powers of the variable factors: \((x-3)^2\) has degree \(2\), and \((x+4)^3\) has degree \(3\). Therefore, the total degree is \(2+3=5\). Answer: B
5.
Solve for the exact value(s) of \(x\): \((x-5)^3+2=-6\).
Subtract 2 from both sides: \((x-5)^3=-8\). Take the cube root: \(x-5=-2\). Therefore, \(x=3\). Answer: D
6.
Which of the following are characteristics of all polynomial curves?
Polynomial curves are smooth and continuous. They do not have sharp corners, and they do not all have a root at the origin. Answer: C
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