1. Classify the following statements as true or false: (a) $-2 \in \mathbb{N}$: False, natural numbers are positive integers starting from 1. (b) $\sqrt{5} \in \mathbb{Q}$: False, $\sqrt{5}$ is irrational. (c) $\pi \in \mathbb{Q}$: False, $\pi$ is irrational. (d) $-1 + \sqrt{-5} \in \mathbb{C}$: True, it is a complex number. (e) 7 is prime: True, 7 has no divisors other than 1 and 7. (f) 9 is prime: False, 9 = 3 \times 3. (g) 3.2 \in \mathbb{Q}$: True, 3.2 = 16/5. (h) $\sqrt{-4} \notin \mathbb{R}$: True, square root of negative number is not real. (i) $\sqrt{125} \in \mathbb{Q}$: False, $\sqrt{125} = 5\sqrt{5}$ is irrational. (j) 4 \in \mathbb{Z}$: True, 4 is an integer. (k) 1 \in \mathbb{C}$: True, 1 is a complex number. (l) 1 is prime: False, 1 is not considered prime. (m) 1.01 \in \mathbb{Q}$: True, 1.01 = 101/100. (n) $\sqrt[3]{125} \in \mathbb{Q}$: True, $\sqrt[3]{125} = 5$. (o) $\sqrt[3]{-27} \in \mathbb{R}$: True, equals -3. (p) $\sqrt{-1} \in \mathbb{R}$: False, equals $i$ which is imaginary. 2. Write the following as fractions: (a) $1.8\̇ = \frac{19}{10}$. (b) $3.32\̇ = \frac{100}{33}$. (c) $8.32\̇3\̇$ repeating 2 digits after decimal: convert carefully using geometric series or algebra. (d) $4.42\̇3\̇$ similar process. (e) $5.0 3\̇2\̇$ repeating digits as fraction. (f) $6. 32\̇33\̇4\̇$ complex repeating decimals. (g) $1. 97\̇6$ (details needed to convert correctly). (h) $2. 34567\̇8\̇$ repeat pattern. (Full expansions require detailed algebraic steps for repeating decimals.) 3. Counterexample that product of two irrationals can be rational: Example: $\sqrt{2} \times \sqrt{2} = 2$ (rational). 4. Classify True/False: - $x$ even $\Rightarrow x^2$ even: True. - $x$ even $\Rightarrow \sqrt{x}$ even: False, $\sqrt{4}=2$ (even), but $\sqrt{16}=4$ (even), but $\sqrt{2}$ is irrational, not integer. - $x$ odd $\Rightarrow \sqrt{x}$ odd: False, as $\sqrt{3}$ is irrational. - $x$ odd $\Rightarrow x^2$ odd: True, product of odd numbers is odd. - $\theta < 180^\circ \Rightarrow \theta$ acute: False, acute angles are $< 90^\circ$. - $x$ even $\Rightarrow x + 1$ odd: True. - $a > b$ $\Rightarrow \frac{1}{a} > \frac{1}{b}$ for $a,b\neq 0$: False, inverses reverse inequalities if both positive. 5. Prove identities: (a) $x(x+1)(x+2) - x(x+2)(x-3) = 4x(x+2)$ Expand both sides and simplify. (b) $(2x+1)^2 - (2x-1)^2 = 8x$ Using difference of squares and binomial expansions. (c) $a^2 + b^2 = (a+b)^2 - 2ab$ (standard algebraic identity). (d) $a^3 + b^3 = (a+b)^3 - 3ab(a+b)$ (standard cubic expansion). (e) $(a-b)^2 = (a+b)^2 - 4ab$ (by expansion). (f) $4x^3 = x^2(x+1)^2 - (x-1)^2 x^2$ Verify by expanding both sides. 6. Simplify expressions: (a) $\frac{2}{x-1} + \frac{3x}{x+2}$ Find common denominator and combine. (b) $\frac{1}{x+2} + \frac{2}{x+2} = \frac{3}{x+2}$. (c) $\frac{1}{x^2-25} - \frac{4}{x+5}$, factor denominator and simplify. (d) $\frac{1}{x^2+5} - \frac{4}{x+5}$, no factor, just combine over common denominator. (e) $\frac{1}{\sqrt{x+1}} + \frac{4}{(x+1)^2} - \frac{1}{x-2}$ (f) $\frac{1}{x^2-16} - \frac{3x}{x-4} + \frac{3}{x+4}$ 7. Factorize: (a) $x^2 - 25 = (x-5)(x+5)$ (b) $x^2 - 49 = (x-7)(x+7)$ (c) $p^2 - 64 = (p-8)(p+8)$ (d) $4x^2 - 9 = (2x - 3)(2x + 3)$ (e) $9x^2 - 4 = (3x - 2)(3x + 2)$ (f) $9 - 25r^2 = (3 - 5r)(3 + 5r)$ (g) $400x^2 - 16y^2 = (20x - 4y)(20x + 4y)$ (h) $169y^2 - 144x^2 = (13y - 12x)(13y + 12x)$ 8. Expand: (a) $(x+a)(x-a) = x^2 - a^2$ (b) $(x-3)(x+7) = x^2 +4x -21$ (c) $(x-3)^2 = x^2 - 6x +9$ (d) $(3a - 5b)(3a + 5b) = 9a^2 - 25b^2$ (e) $(7 - \sqrt{2})(7 + \sqrt{2}) = 49 - 2 = 47$ (f) $(4\sqrt{x} + 5y)(4\sqrt{x} - 5y) = 16x - 25y^2$ 9. Solve quadratics (with discriminants): (a) $(x-5)(x+2)=0$ roots: $x=5,-2$ real and distinct. (b) $5x^2 -17x +6=0$ use quadratic formula. (c) $x^2 -2x =7$ rewrite as $x^2 -2x -7=0$ solve. (d) $x(x+10)=-25$ rewrite $x^2 +10x +25=0$ discriminant zero, one root. (e) $4x^2 -3x=0$ factor. (f) $2x^2 -5x -1=0$ quadratic formula. (g) $9x^2 -1=0$ roots. (h) $x^2 -4x +4=0$ perfect square. 10. Inequalities: (a) $2x > 5 \Rightarrow x > \frac{5}{2}$. (b) $3x -1 >9 \Rightarrow 3x >10 \Rightarrow x > \frac{10}{3}$. (c) $2x -5 <0 \Rightarrow x < \frac{5}{2}$. (d) $-\frac{6x}{5} \geq 13 \Rightarrow -6x \geq 65 \Rightarrow x \leq -\frac{65}{6}$. (e) $\frac{x}{2} - \frac{5x}{3} < 0 \Rightarrow -\frac{7x}{6} < 0 \Rightarrow x > 0$. (f) $x^2 > -1$ always true for all real $x$. 17. Long division examples: (a) Divide $x^2 + 2x + 5$ by $x + 2$ using polynomial division. Divide highest degree terms, subtract, bring down next term. Repeat until degree of remainder less than divisor. (b) Similarly for other divisions applying polynomial long division rules. Slug: worksheet classification Subject: algebra Desmos: {"latex":"","features":{"intercepts":true,"extrema":true}} q_count: 4