1.
Factor completely. \(x^3+8x^2-7x\).
Factor out \(x\): \(x(x^2+8x-7)\). The quadratic does not factor over the integers. Answer: B
2.
Factor completely. \(15x^2-27x-6\).
Factor out 3 then factor: \(15x^2-27x-6=3(5x^2-9x-2)=3(x-2)(5x+1)\). Answer: B
3.
Determine the remainder when \(P(x)=9x^3+26x^2-12x-1\) is divided by \(3x-1\).
Set \(x=\frac13\). \(P(\frac13)=\frac13+\frac{6}{9}-4-1=-4\), which is answer A
4.
Find the remainder when \(P(x)=-5x^3-23x^2-23x-23\) is divided by \(x+3\).
Remainder Theorem: evaluate \(P(-3)=135-207+69-23=-26\). Answer: B (the provided key marks D, but the computed remainder is \(-26\)).
5.
Solve for \(x\): \(6x^2-17x+5=0\).
Factor: \((3x-1)(2x-5)=0\). Thus \(x=\frac13,\frac52\). Answer: D
6.
Factor completely. \(1-9x^2\).
Difference of squares: \(1-(3x)^2=(1-3x)(1+3x)\). Answer: D
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