In Exercises polyarithexfirst - polyarithexlast, perform the indicated operations and simplify.
\((4-3x) + (3x^2 + 2x + 7)\)

\(3x^2 - x + 11\)
\(t^2 + 4t - 2(3-t)\)

\(t^2 + 6t-6\)
\(q(200-3q) - (5q + 500)\)

\(-3q^2+195q-500\)
\((3y-1)(2y+1)\)

\(6y^2+y-1\)
\(\left (3-\frac {x}{2}\right )(2x+5)\)

\(-x^2 + \frac {7}{2} x + 15\)
\(-(4t+3)(t^2-2)\)

\(-4t^3-3t^2+8t+6\)
\(2w(w^3-5)(w^3+5)\)

\(2w^7 - 50w\)
\((5a^2 - 3)(25a^4 + 15a^2 + 9)\)

\(125a^6 - 27\)
\((x^2-2x+3)(x^2+2x+3)\)

\(x^4+2x^2+9\)
\((\sqrt {7} - z)(\sqrt {7} + z)\)

\(7-z^2\)
\((x - \sqrt [3]{5})^3\)

\(x^3 - 3x^2\sqrt [3]{5} + 3x\sqrt [3]{25} - 5\)
\((x - \sqrt [3]{5})(x^2 + x\sqrt [3]{5} + \sqrt [3]{25})\)

\(x^3 - 5\)
\((w-3)^2 - (w^2 + 9)\)

\(-6w\)
\((x+h)^2 - 2(x+h) - (x^2 - 2x)\)

\(h^2 + 2xh - 2h\)
\((x-[2+\sqrt {5}])(x-[2-\sqrt {5}])\)

\(x^2 - 4x - 1\)
In these Problems, perform the indicated division. Check your answer by showing
\[\text {dividend} = (\text {divisor})( \text {quotient}) + \text {remainder}\]
\( (5x^2 - 3x + 1) \div (x + 1) \)

quotient: \(\answer {5}x + \answer {-8}\)

remainder: \(\answer {9}\)

\((3y^2 + 6y - 7) \div (y-3)\)

quotient: \(\answer {3}y + \answer {15}\)

remainder: \(\answer {38}\)

\((6w - 3) \div (2w+5)\)

quotient: \(\answer {3}\)

remainder: \(\answer {18}\)

\((2x+1) \div (3x-4)\)

quotient: \(\answer {\frac {2}{3}}\)

remainder: \(\answer {\frac {11}{3}}\)

\((t^2 - 4) \div (2t + 1)\)

quotient: \(\frac {t}{2} - \frac {1}{4}\)

remainder: \(-\frac {15}{4}\)

\((w^3 - 8) \div (5w-10)\)

quotient: \(\frac {w^2}{5} + \frac {2w}{5} + \frac {4}{5}\)

remainder: \(0\)

\((2x^2 - x + 1) \div (3x^2 + 1)\)

quotient: \(\frac {2}{3}\)

remainder: \(-x + \frac {1}{3}\)

\((4y^4+3y^2+1) \div (2y^2-y+1)\)

quotient: \(2y^2+y+1\)

remainder: \(0\)

\(w^4 \div (w^3 - 2)\)

quotient: \(w\)

remainder: \(2w\)

\((5t^3 - t + 1) \div (t^2 + 4)\)

quotient: \(5t\)

remainder: \(-21t + 1\)

\((t^3 - 4) \div (t - \sqrt [3]{4})\)

quotient: \(t^2 + t \sqrt [3]{4} + 2\sqrt [3]{2}\)

remainder: \(0\)

\((x^2-2x-1) \div (x-[1-\sqrt {2}])\)

quotient: \(x -1 - \sqrt {2}\)

remainder: 0

In Exercises specialformexfirst - specialformexlast verify the given formula by showing the left hand side of the equation simplifies to the right hand side of the equation.
Perfect Cube: \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)
Difference of Cubes: \((a - b)(a^2 + ab + b^2) = a^3 - b^3\)
Sum of Cubes: \((a + b)(a^2 - ab + b^2) = a^3 + b^3\)
Perfect Quartic: \((a+b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4\)
Difference of Quartics: \((a-b)(a+b)(a^2+b^2) = a^4 - b^4\)
Sum of Quartics: \((a^2 + ab \sqrt {2} + b^2)(a^2 - ab \sqrt {2} + b^2) = a^4 + b^4\)
With help from your classmates, determine under what conditions \((a+b)^2 = a^2 + b^2\). What about \((a+b)^3 = a^3 + b^3\)? In general, when does \((a+b)^n = a^n + b^n\) for a natural number \(n \geq 2\)?