1.
Factor completely. \(3x^3-3x\).
Factor out the GCF \(3x\): \(3x^3-3x=3x(x^2-1)\). Then use difference of squares: \(x^2-1=(x-1)(x+1)\). Therefore, \(3x(x-1)(x+1)\). Answer: C
2.
When factored completely, one of the factors of \(2x^2-xy-15y^2\) is:
Factor by grouping: \(2x^2-xy-15y^2=2x^2-6xy+5xy-15y^2=2x(x-3y)+5y(x-3y)=(x-3y)(2x+5y)\). One factor is \(2x+5y\). Answer: D
3.
Factor completely. \(x^2-17x+60\).
Find two numbers with product \(60\) and sum \(-17\): \(-5\) and \(-12\). Therefore, \(x^2-17x+60=(x-5)(x-12)\). Answer: C
4.
Solve for \(x\): \(3x^2-8x=0\).
Factor out \(x\): \(3x^2-8x=x(3x-8)=0\). Therefore, \(x=0\) or \(3x-8=0\Rightarrow x=\frac83\). Answer: C
5.
Factor: \(5y^2+30y-xy-6x\).
Factor by grouping: \(5y^2+30y-xy-6x=5y(y+6)-x(y+6)=(y+6)(5y-x)\). Answer: D
6.
Which of the following are the roots of the factored equation \((2x+1)(x-3)=0\)?
Set each factor equal to zero: \(2x+1=0\Rightarrow x=-\frac12\), and \(x-3=0\Rightarrow x=3\). Answer: B
7.
Factor completely. \(24x^2-40x+16\).
Factor out \(8\): \(24x^2-40x+16=8(3x^2-5x+2)\). Then \(3x^2-5x+2=(x-1)(3x-2)\). Therefore, \(8(x-1)(3x-2)\). Answer: C
8.
Solve for \(x\): \(x(x+1)-3(x+2)=2\).
Expand and simplify: \(x(x+1)-3(x+2)=2\Rightarrow x^2+x-3x-6=2\Rightarrow x^2-2x-8=0\). Factor: \((x+2)(x-4)=0\). Thus \(x=-2\) or \(x=4\). Answer: A
9.
Solve for \(x\): \(12x^2-5x-2=0\).
Factor: \(12x^2-5x-2=(4x+1)(3x-2)\). Set each factor equal to zero: \(4x+1=0\Rightarrow x=-\frac14\), and \(3x-2=0\Rightarrow x=\frac23\). Answer: C
10.
What is the maximum number of turns and the maximum number of \(x\)-intercepts that a polynomial of degree \(7\) can have?
An \(n\)th-degree polynomial can have at most \(n-1\) turns and at most \(n\) \(x\)-intercepts. For degree \(7\), the maximum number of turns is \(6\), and the maximum number of \(x\)-intercepts is \(7\). Answer: B
11.
Which of the following does not have \((2x-1)\) as a factor?
Use the Factor Theorem with \(x=\frac12\). For \(x^3-2x^2+5\), \(P(\frac12)=\frac18-2(\frac14)+5=\frac18-\frac12+5=\frac{37}{8}\neq0\). Therefore, \((2x-1)\) is not a factor. Answer: E
12.
Factor completely: \(P(x)=4x^3-4x^2-7x-2\).
Test \(x=2\): \(4(2)^3-4(2)^2-7(2)-2=32-16-14-2=0\), so \((x-2)\) is a factor. Dividing gives \(4x^2+4x+1=(2x+1)^2\). Therefore, \(P(x)=(x-2)(2x+1)^2\). Answer: C
13.
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: D
14.
Which of the following best describes the graph of a polynomial of degree \(6\) with a negative leading coefficient?
A polynomial with even degree has both ends pointing in the same direction. A negative leading coefficient means both ends point down. Answer: D
15.
Determine the equation of the cubic polynomial graphed below.
From the graph, the roots are \(x=-2\), \(x=1\), and \(x=3\). So the polynomial is \((x+2)(x-1)(x-3)\). Expanding gives \((x+2)(x^2-4x+3)=x^3-2x^2-5x+6\). Answer: C
16.
Determine the division statement for \((5x^3+23x^2-42x+16)\div(x+6)\).
Divide \(5x^3+23x^2-42x+16\) by \(x+6\). The quotient is \(5x^2-7x\) and the remainder is \(16\). Therefore, the division statement is \((5x^2-7x)(x+6)+16\). Answer: A
17.
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: B
18.
Determine the root with highest multiplicity in the polynomial \(y=7(x-4)^2(x+3)\).
The factor \((x-4)^2\) gives the root \(x=4\) with multiplicity \(2\). The factor \((x+3)\) gives the root \(x=-3\) with multiplicity \(1\). Therefore, the root with highest multiplicity is \(x=4\). Answer: D
19.
Which of the following is a polynomial curve?
A polynomial curve must be smooth and continuous. Among the given graphs, Graph B is smooth and continuous. Answer: B
20.
Solve by factoring: \(x^3+4x^2-7x-10=0\).
Test \(x=-1\): \((-1)^3+4(-1)^2-7(-1)-10=-1+4+7-10=0\), so \((x+1)\) is a factor. Dividing gives \(x^2+3x-10=(x+5)(x-2)\). Thus the roots are \(x=-5,-1,2\). Answer: D
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