1.
Solve for \(d\): \(d^2=21-4d\).
Move all terms to one side: \(d^2+4d-21=0\). Factor: \((d-3)(d+7)=0\). Therefore, \(d=3\) or \(d=-7\). Answer: C
2.
Factor completely. \(32x^2-50y^2\).
Factor out 2, then use difference of squares: \(2[(4x)^2-(5y)^2]=2(4x-5y)(4x+5y)\). Answer: A
3.
Find the remainder when \(P(x)=2x^3-13x^2+11x+26\) is divided by \(x-5\).
Using the Remainder Theorem, for divisor \(x-5\), evaluate \(P(5)\). \(P(5)=2(5)^3-13(5)^2+11(5)+26=250-325+55+26=6\). Answer: B
4.
Determine the remainder when \(P(x)=9x^3+6x^2-12x-1\) is divided by \(3x-1\).
For divisor \(3x-1\), set \(3x-1=0\), so \(x=\frac13\). Evaluate \(P\!\left(\frac13\right)=9\left(\frac13\right)^3+6\left(\frac13\right)^2-12\left(\frac13\right)-1=\frac13+\frac23-4-1=-4\). Answer: A
5.
Factor completely. \(-x^3+5x^2+10x\).
Factor out the GCF: \(-x(x^2-5x-10)\). The quadratic cannot be factored further over the integers. Answer: C
6.
Factor completely. \(10x^3+25x^2-15x\).
Factor out \(5x\), then factor \(2x^2+5x-3=(2x-1)(x+3)\). Answer: B
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