Question

simplify (8pts each)

1. $\frac{k^2 - 11k + 30}{k - 5}$
2. $\frac{9n^2 - 36n}{n - 4}$
3. $\frac{x^2 + 7x + 6}{9x + 54}$
4. $\frac{5k + 35}{k - 5} cdot \frac{k - 5}{5}$
5. $\frac{6}{n + 7} cdot \frac{n^2 + 6n - 7}{n - 1}$
6. $\frac{v - 4}{v - 2} div \frac{v + 8}{v^2 + 6v - 16}$
7. $\frac{2}{k + 3} div \frac{k + 2}{k^2 + 5k + 6}$

bonus (+5pts)

1. factor completely

$4x^2 - 44x + 96$

Explanation:

Problem 9

Step1: Factor numerator

$k^2 -11k +30 = (k-5)(k-6)$

Step2: Cancel common factors

$\frac{(k-5)(k-6)}{k-5} = k-6$ (where $k
eq 5$)

Problem 10

Step1: Factor numerator

$9n^2 -36n = 9n(n-4)$

Step2: Cancel common factors

$\frac{9n(n-4)}{n-4} = 9n$ (where $n
eq 4$)

Problem 11

Step1: Factor numerator and denominator

$x^2+7x+6=(x+1)(x+6)$; $9x+54=9(x+6)$

Step2: Cancel common factors

$\frac{(x+1)(x+6)}{9(x+6)} = \frac{x+1}{9}$ (where $x
eq -6$)

Problem 12

Step1: Factor numerator of first term

$5k+35=5(k+7)$

Step2: Multiply and cancel factors

$\frac{5(k+7)}{k-5} \cdot \frac{k-5}{5} = k+7$ (where $k
eq 5$)

Problem 13

Step1: Factor numerator of second term

$n^2+6n-7=(n+7)(n-1)$

Step2: Multiply and cancel factors

$\frac{6}{n+7} \cdot \frac{(n+7)(n-1)}{n-1} = 6$ (where $n
eq -7, 1$)

Problem 14

Step1: Rewrite division as multiplication

$\frac{v-4}{v-2} \cdot \frac{v^2+6v-16}{v+8}$

Step2: Factor quadratic trinomial

$v^2+6v-16=(v+8)(v-2)$

Step3: Multiply and cancel factors

$\frac{v-4}{v-2} \cdot \frac{(v+8)(v-2)}{v+8} = v-4$ (where $v
eq 2, -8$)

Problem 15

Step1: Rewrite division as multiplication

$\frac{2}{k+3} \cdot \frac{k^2+5k+6}{k+2}$

Step2: Factor quadratic trinomial

$k^2+5k+6=(k+2)(k+3)$

Step3: Multiply and cancel factors

$\frac{2}{k+3} \cdot \frac{(k+2)(k+3)}{k+2} = 2$ (where $k
eq -3, -2$)

Problem 16 (Bonus)

Step1: Factor out GCF

$4x^2-44x+96=4(x^2-11x+24)$

Step2: Factor quadratic trinomial

$x^2-11x+24=(x-3)(x-8)$

Step3: Write fully factored form

$4(x-3)(x-8)$

Answer:

1. $k-6$ ($k

eq 5$)

1. $9n$ ($n

eq 4$)

1. $\frac{x+1}{9}$ ($x

eq -6$)

1. $k+7$ ($k

eq 5$)

1. $6$ ($n

eq -7, 1$)

1. $v-4$ ($v

eq 2, -8$)

1. $2$ ($k

eq -3, -2$)

1. $4(x-3)(x-8)$