1.
Factor completely. \(6x^2+12x-48\).
Factor out \(6\): \(6(x^2+2x-8)\). Then factor the quadratic: \(x^2+2x-8=(x+4)(x-2)\). So the complete factorization is \(6(x+4)(x-2)\). Answer: A
2.
Factor completely. \(100x^2y^2-169z^2\).
Use difference of squares: \(100x^2y^2-169z^2=(10xy)^2-(13z)^2=(10xy-13z)(10xy+13z)\). Answer: E
3.
Find the remainder when \(P(x)=4x^3-11x^2+13x+8\) is divided by \(x-1\).
Using the Remainder Theorem, for divisor \(x-1\), evaluate \(P(1)\). \(P(1)=4(1)^3-11(1)^2+13(1)+8=4-11+13+8=14\). Answer: B
4.
Solve for \(m\): \(m^2-m=42\).
Move all terms to one side: \(m^2-m-42=0\). Factor: \((m+6)(m-7)=0\). Thus \(m=-6\) or \(m=7\). Answer: B
5.
Determine the remainder when \(P(x)=4x^3-6x^2+4x-3\) is divided by \(2x-1\).
For divisor \(2x-1\), set \(2x-1=0\), so \(x=\frac12\). Evaluate \(P\!\left(\frac12\right)=4\left(\frac12\right)^3-6\left(\frac12\right)^2+4\left(\frac12\right)-3=\frac12-\frac32+2-3=-2\). Answer: B
6.
Factor completely. \(16x^2-16x+4\).
Factor out \(4\): \(16x^2-16x+4=4(4x^2-4x+1)\). The trinomial is a perfect square: \(4x^2-4x+1=(2x-1)^2\). Therefore the answer is \(4(2x-1)^2\). Answer: C
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