Exam 2: Introduction to Algebra

Square this binomial: $\left(a^{4}-b^{n}\right)$

Divide the polynomial by the trinomial: $10(a-b)^{2}+34(a-b)+12$ by $5(a-b)+2$

Express without fractions, using negative exponents where needed: $\frac{a^{-2} b^{3}}{c^{4}}$

Multiply these binomials, and simplify: $(5 a+2 b)(-3 a+b)$

Multiply these binomials, and simplify: $(5 p-1)(p+1)(p-2)(p+2)$

Write with positive exponents only: $\left(\frac{-2 x}{y}\right)^{-2}$

Nanoparticles are ideal for chemical reactions because they have a high surface-area-to-volume ratio. The surface-area-to-volume ratio for platinum-nickel nanoparticles in a hydrogen fuel cell is given by $\frac{3.464 x^{2}}{0.4714 x^{3}}$ , where $x$ is the side length. Simplify this ratio.

Divide the polynomial by the monomial: $21 a^{5}-35 a^{2}$ by $7 a^{2}$

Remove the symbols of grouping (i.e., brackets, braces, parentheses) and collect terms: $3(x-y)+\{5 y-$ $[3 x-(2 x-y)]+4(x-y)\}$

Divide the polynomial by the trinomial: $x^{6}+15 x^{4}+24 x^{2}-144$ by $x^{4}+3 x^{2}-12$

Write with positive exponents only: $3 x^{-5} y+12 y^{-4}$

Multiply the binomial and trinomial, and simplify: $(a-2 b)(3 a+5 b+6)$

Divide the polynomial by the binomial: $a^{3} c^{6}+a^{2} c^{4}-a c^{2}-1$ by $a c^{2}+1$

Multiply these binomials, and simplify: $(c-3)(c+5)(c+5)$

Multiply the polynomial by the monomial, and simplify: $-15 x y^{2}\left(y^{3}-2 x y+3 x^{2}-x^{2} y\right)$

The surface area of a cylinder with radius $r$ and perpendicular height $3 r$ is given by $2 \pi\left(r^{2}+3 r^{2}\right)$ . Simplify this expression, keeping $\pi$ in your answer.

Multiply these trinomials, and simplify: $(2 x-3 y+5 z)(-x+2 y-4 z)$

Remove the symbols of grouping (i.e., brackets, braces, parentheses) and collect terms: $\frac{3}{4}(x-2)$ - $\frac{1}{2}\left(x^{2}+3\right)$

The volume of a cone of radius $2 r$ and perpendicular height $3 r$ is given by $\frac{\pi}{3}(2 r)^{2}(3 r)$ . Simplify this expression.

Multiply these monomials, and simplify: $a^{4}(a)$