1. **Problem 3: Factor using the difference of squares** The difference of squares formula is: $$a^2 - b^2 = (a - b)(a + b)$$ We apply this to each expression: a) $a^2 - 1 = (a - 1)(a + 1)$ b) $b^2 - 4 = (b - 2)(b + 2)$ c) $c^2 - 9 = (c - 3)(c + 3)$ d) $d^2 - 100 = (d - 10)(d + 10)$ e) $25 - y^2 = (5 - y)(5 + y)$ f) $1 - n^2 = (1 - n)(1 + n)$ g) $49 - x^2 = (7 - x)(7 + x)$ h) $144 - p^2 = (12 - p)(12 + p)$ i) $4c^2 - 9 = (2c - 3)(2c + 3)$ j) $9u^2 - 1 = (3u - 1)(3u + 1)$ k) $25x^2 - 16 = (5x - 4)(5x + 4)$ l) $1 - 49k^2 = (1 - 7k)(1 + 7k)$ m) $x^2 - 4y^2 = (x - 2y)(x + 2y)$ n) $9a^2 - b^2 = (3a - b)(3a + b)$ o) $25m^2 - 36n^2 = (5m - 6n)(5m + 6n)$ p) $81a^2b^2 - 64 = (9ab - 8)(9ab + 8)$ 2. **Problem 4: Factor each monic quadratic expression** For monic quadratics $x^2 + bx + c$, find two numbers that multiply to $c$ and add to $b$. a) $a^2 + 3a + 2$ - Factors of 2 that add to 3: 1 and 2 - Factorization: $(a + 1)(a + 2)$ b) $k^2 + 5k + 6$ - Factors of 6 that add to 5: 2 and 3 - Factorization: $(k + 2)(k + 3)$ c) $m^2 + 7m + 6$ - Factors of 6 that add to 7: 1 and 6 - Factorization: $(m + 1)(m + 6)$ d) $x^2 + 8x + 15$ - Factors of 15 that add to 8: 3 and 5 - Factorization: $(x + 3)(x + 5)$ e) $y^2 + 9y + 20$ - Factors of 20 that add to 9: 4 and 5 - Factorization: $(y + 4)(y + 5)$ f) $t^2 + 12t + 20$ - Factors of 20 that add to 12: 2 and 10 - Factorization: $(t + 2)(t + 10)$ g) $x^2 - 4x + 3$ - Factors of 3 that add to -4: -1 and -3 - Factorization: $(x - 1)(x - 3)$ h) $c^2 - 7c + 10$ - Factors of 10 that add to -7: -2 and -5 - Factorization: $(c - 2)(c - 5)$ i) $a^2 - 7a + 12$ - Factors of 12 that add to -7: -3 and -4 - Factorization: $(a - 3)(a - 4)$ j) $b^2 - 8b + 12$ - Factors of 12 that add to -8: -2 and -6 - Factorization: $(b - 2)(b - 6)$ k) $t^2 + 1 - 2$ - Rewrite as $t^2 - 1$ - Difference of squares: $(t - 1)(t + 1)$ l) $u^2 - u - 2$ - Factors of -2 that add to -1: -2 and 1 - Factorization: $(u - 2)(u + 1)$ m) $w^2 - 2w - 8$ - Factors of -8 that add to -2: -4 and 2 - Factorization: $(w - 4)(w + 2)$ n) $a^2 + 2a - 8$ - Factors of -8 that add to 2: 4 and -2 - Factorization: $(a + 4)(a - 2)$ o) $p^2 - 2p - 15$ - Factors of -15 that add to -2: -5 and 3 - Factorization: $(p - 5)(p + 3)$ p) $y^2 + 3y - 28$ - Factors of -28 that add to 3: 7 and -4 - Factorization: $(y + 7)(y - 4)$ q) $c^2 - 12c + 27$ - Factors of 27 that add to -12: -9 and -3 - Factorization: $(c - 9)(c - 3)$ r) $u^2 - 13u + 42$ - Factors of 42 that add to -13: -6 and -7 - Factorization: $(u - 6)(u - 7)$ s) $x^2 - x - 90$ - Factors of -90 that add to -1: -10 and 9 - Factorization: $(x - 10)(x + 9)$ t) $x^2 + 3x - 40$ - Factors of -40 that add to 3: 8 and -5 - Factorization: $(x + 8)(x - 5)$ u) $t^2 - 4t - 32$ - Factors of -32 that add to -4: -8 and 4 - Factorization: $(t - 8)(t + 4)$ v) $p^2 + 9p - 36$ - Factors of -36 that add to 9: 12 and -3 - Factorization: $(p + 12)(p - 3)$ w) $u^2 - 16u - 80$ - Factors of -80 that add to -16: -20 and 4 - Factorization: $(u - 20)(u + 4)$ x) $r^2 + 23r - 50$ - Factors of -50 that add to 23: 25 and -2 - Factorization: $(r + 25)(r - 2)$