1.
Factor completely. \(x^2-14x+13\).
Find two numbers with product \(13\) and sum \(-14\): \(-1\) and \(-13\). Therefore, \(x^2-14x+13=(x-1)(x-13)\). Answer: A
2.
Factor completely. \(144x^2y^2-121z^2\).
Use difference of squares: \(144x^2y^2-121z^2=(12xy)^2-(11z)^2=(12xy-11z)(12xy+11z)\). Answer: C
3.
Solve for \(x\): \(\frac{x+10}{x-8}=x\).
Multiply by \(x-8\): \(x+10=x(x-8)\). Then \(0=x^2-9x-10=(x+1)(x-10)\). Therefore, \(x=-1\) or \(x=10\). Answer: A
4.
When factored completely, one of the factors of \(2x^2-xy-15y^2\) is:
Factor by grouping: \(2x^2-xy-15y^2=2x^2-6xy+5xy-15y^2=(x-3y)(2x+5y)\). One factor is \(2x+5y\). Answer: A
5.
Solve for \(x\): \(5x^2-5x=0\).
Factor: \(5x^2-5x=5x(x-1)=0\). Therefore, \(x=0\) or \(x=1\). Answer: A
6.
Find the remainder when \(P(x)=-5x^3-23x^2-23x-23\) is divided by \(x+3\).
Using the Remainder Theorem, for divisor \(x+3\), evaluate \(P(-3)\). \(P(-3)=-5(-3)^3-23(-3)^2-23(-3)-23=135-207+69-23=-26\). Answer: B
7.
Solve for \(t\): \(t^2-49=0\).
\(t^2-49=0\Rightarrow t^2=49\Rightarrow t=\pm7\). Answer: C
8.
Solve \((6x^3-19x^2+19x-19)\div(3x-2)\).
Polynomial division gives quotient \(2x^2-5x+3\). Multiplying back: \((3x-2)(2x^2-5x+3)=6x^3-19x^2+19x-6\), so the remainder is \(-13\). Answer: B
9.
Factor: \(c^2+6c+9-36d^2\).
First factor the trinomial: \(c^2+6c+9=(c+3)^2\). Then use difference of squares: \((c+3)^2-(6d)^2=(c+3+6d)(c+3-6d)\). Answer: A
10.
What are the maximum and minimum numbers of real roots that a polynomial of degree \(5\) can have?
A degree \(5\) polynomial can have at most \(5\) real roots. Since degree \(5\) is odd, it must have at least \(1\) real root. Answer: A
11.
Which of the following is a polynomial curve?
A polynomial curve must be smooth and continuous. Among the given graphs, Graph A is smooth and continuous. Answer: A
12.
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: B
13.
Solve for \(x\): \(P(x)=x^3-8x^2+x+42=0\).
Test \(x=-2\): \((-2)^3-8(-2)^2+(-2)+42=-8-32-2+42=0\), so \((x+2)\) is a factor. Dividing gives \(x^2-10x+21=(x-7)(x-3)\). Roots are \(-2,3,7\). Answer: D
14.
Factor completely: \(P(x)=6x^3-x^2-4x-1\).
Test \(x=1\): \(6-1-4-1=0\), so \((x-1)\) is a factor. Dividing gives \(6x^2+5x+1=(2x+1)(3x+1)\). Therefore, \(P(x)=(2x+1)(3x+1)(x-1)\). Answer: A
15.
Solve for the exact value(s) of \(x\): \((x-5)^3+2=-6\).
Subtract \(2\): \((x-5)^3=-8\). Take the cube root: \(x-5=-2\). Therefore, \(x=3\). Answer: D
16.
Factor completely. \(10x^3+25x^2-15x\).
Factor out \(5x\): \(10x^3+25x^2-15x=5x(2x^2+5x-3)\). Then \(2x^2+5x-3=(x+3)(2x-1)\). Answer: B
17.
A 3rd degree polynomial graph has a root with a multiplicity of \(2\). The function has another root; what is its multiplicity?
The sum of the multiplicities must equal the degree. Since the degree is \(3\) and one root has multiplicity \(2\), the other root has multiplicity \(3-2=1\). Answer: B
18.
Which of the following best describes the graph of a polynomial of degree \(6\) with a negative leading coefficient?
An even degree polynomial has both ends pointing in the same direction. A negative leading coefficient means both ends point down. Answer: A
19.
Which graph is a possible representation of \(y=ax^4+bx^3+cx-6\), where \(a
Since the degree is \(4\) and \(a<0\), both ends of the graph must point down. The constant term is \(-6\), so the \(y\)-intercept is \(-6\). Graph D matches these features. Answer: D
20.
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: A
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