Practice Test 35 - Algebra 2

20 questions. Answer them in any order, then submit to see your score and every explanation.

For the polynomial $T(x) = 2x^4 - 3x^3 + x^2 - 4x + 5$ , suppose that when $T(x)$ is divided by $(x - a)$ , the remainder is $r$ . What is the remainder when $T(2x)$ is divided by $(x - \frac{a}{2})$ ?
Which restriction(s) make $f(x)=x^2-4x$ pass the horizontal line test and therefore have an inverse? (Choose the best answer.)
The sequence is defined recursively by $a_1=6$ and $a_{n+1}=\tfrac{1}{2}a_n$ . Write the explicit formula for $a_n$ (treating it as a function with integer domain $n\ge 1$ ).
Factor and sketch $f(x)=x^4-5x^2+4$ showing all real zeros and the end behavior.
Given $t(x) = \frac{2x + 1}{x - 3}$ where $x \neq 3$ , if $t^{-1}(x)$ exists, what is the domain of $t^{-1}(x)$ ?
A school club is making gift bags using stickers and pencils. Let $x$ be the number of sticker packs and $y$ be the number of pencil packs purchased. Each sticker pack costs $4 and each pencil pack costs $6. The club has at most $72 to spend, needs at least 10 total packs, and cannot buy negative amounts. Which system of inequalities represents all constraints?
Use the identity $a^2-b^2=(a+b)(a-b)$ , which remains valid for complex $a,b$ , to evaluate $(1+i)^2-(2-i)^2.$
Rationalize the denominator using the conjugate and write in $a+bi$ form: $\frac{2-i}{3+2i}.$
What is the 4th term (counting from the first term as term 1) in the expansion of $(x+y)^6?$
Compute the product using the conjugate pair identity $(a+bi)(a-bi)=a^2+b^2$ : $(4+3i)(4-3i).$ . Give your answer in simplest form.
A quadratic with no real zeros can still factor over $\mathbb{C}$ . Which option correctly factors $P(x)=x^2+2x+5$ over the complex numbers?
A geometric series is the sum of terms of a geometric sequence. For the finite geometric series $S_n=a+ar+ar^2+\cdots+ar^{n-1},$ derive a formula for $S_n$ (assume $r\ne 1$ ) by multiplying by $r$ and subtracting so that the middle terms cancel.
A ball is thrown upward from a height of 6 feet. The function $h(t) = -16t^2 + 32t + 6$ models the height of the ball in feet after $t$ seconds. Which statement correctly interprets a key feature of this function in the context of the problem?
Let the parent function be $f(x)=\sqrt{x}$ . Describe the effect of replacing $f(x)$ with $g(x)=f(2x)$ .
Use synthetic division (a streamlined alternative to long division for $(x-c)$ ) to divide $P(x)=x^3-6x^2+11x-6$ by $(x-3)$ . Use $c=3$ and coefficients $1,-6,11,-6$ . The bottom row gives the quotient coefficients and the remainder, and the remainder equals $P(3)$ by the Remainder Theorem. What is the quotient and remainder?
What is the coefficient of $x^3y^3$ in the expansion of $(x+y)^6?$ (You may use Pascal's Triangle.)
Use technology (for a $3\times 3$ matrix) to solve the system by writing it as $AX=B$ and using $X=A^{-1}B$ . \begin{align*} x+y+z&=6\\ 2x-y+z&=3\\ x+2y-z&=2 \end{align*} What is $(x,y,z)$ ?
Consider $F(x)=x^2(x+1)-4(x+1).$ If you view $(x+1)$ as a single factor, which statement best describes the structure revealed by chunking?
Derive the finite geometric series sum formula for $r\ne 1$ . Start with $S_n=a+ar+ar^2+\cdots+ar^{n-1}.$ Multiply both sides by $r$ and subtract to eliminate the middle terms. Which expression correctly gives $S_n$ ?
Use the identity $(x^2+y^2)^2=(x^2-y^2)^2+(2xy)^2$ to generate a Pythagorean triple when $x=3$ and $y=2$ . (A Pythagorean triple is $(a,b,c)$ with $a^2+b^2=c^2$ .)

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