Worksheet for simplifying rational expressions with polynomial fractions.

Let’s simplify each rational expression one by one. We’ll factor the numerator and denominator, then cancel any common factors.

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1. $\frac{n^2 + 7n + 6}{n^2 - n - 42}$

Factor numerator:
$n^2 + 7n + 6 = (n + 1)(n + 6)$

Factor denominator:
$n^2 - n - 42 = (n - 7)(n + 6)$

Cancel $(n + 6)$:
$\frac{(n + 1)\cancel{(n + 6)}}{(n - 7)\cancel{(n + 6)}} = \frac{n + 1}{n - 7}$

✔ Final for #1: $\frac{n + 1}{n - 7}$

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2. $\frac{h^2 + 13h + 42}{h + 7}$

Factor numerator:
$h^2 + 13h + 42 = (h + 6)(h + 7)$

Denominator is already $h + 7$

Cancel $(h + 7)$:
$\frac{(h + 6)\cancel{(h + 7)}}{\cancel{h + 7}} = h + 6$

✔ Final for #2: $h + 6$

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3. $\frac{s^2 - 4s - 45}{s^2 + 2s - 15}$

Factor numerator:
$s^2 - 4s - 45 = (s - 9)(s + 5)$

Factor denominator:
$s^2 + 2s - 15 = (s + 5)(s - 3)$

Cancel $(s + 5)$:
$\frac{(s - 9)\cancel{(s + 5)}}{\cancel{(s + 5)}(s - 3)} = \frac{s - 9}{s - 3}$

✔ Final for #3: $\frac{s - 9}{s - 3}$

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4. $\frac{g + 9}{g^2 + 16g + 63}$

Factor denominator:
$g^2 + 16g + 63 = (g + 7)(g + 9)$

Numerator is $g + 9$

Cancel $(g + 9)$:
$\frac{\cancel{g + 9}}{(g + 7)\cancel{(g + 9)}} = \frac{1}{g + 7}$

✔ Final for #4: $\frac{1}{g + 7}$

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5. $\frac{c^2 - 8c + 15}{c - 3}$

Factor numerator:
$c^2 - 8c + 15 = (c - 3)(c - 5)$

Denominator is $c - 3$

Cancel $(c - 3)$:
$\frac{\cancel{(c - 3)}(c - 5)}{\cancel{c - 3}} = c - 5$

✔ Final for #5: $c - 5$

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6. $\frac{8h^2 + 15h - 2}{4h^2 + 11h + 6}$

Factor numerator:
We need two numbers that multiply to $8 \cdot (-2) = -16$, add to 15 → 16 and -1
So: $8h^2 + 16h - h - 2 = 8h(h + 2) -1(h + 2) = (8h - 1)(h + 2)$

Factor denominator:
$4h^2 + 11h + 6$ → Multiply 4·6=24, find factors of 24 that add to 11 → 8 and 3
So: $4h^2 + 8h + 3h + 6 = 4h(h + 2) + 3(h + 2) = (4h + 3)(h + 2)$

Now:
$\frac{(8h - 1)\cancel{(h + 2)}}{(4h + 3)\cancel{(h + 2)}} = \frac{8h - 1}{4h + 3}$

✔ Final for #6: $\frac{8h - 1}{4h + 3}$

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7. $\frac{3r^2 - 4r - 6}{5r^2 - 16r + 3}$

Check if numerator factors:
Discriminant: $(-4)^2 - 4(3)(-6) = 16 + 72 = 88$ → not a perfect square → doesn’t factor nicely over integers.

Denominator: $5r^2 - 16r + 3$
Discriminant: $(-16)^2 - 4(5)(3) = 256 - 60 = 196 = 14^2$ → factors!

Find roots: $r = \frac{16 \pm 14}{10} → r = 3 or r = 0.2 = \frac{1}{5}$

So: $5r^2 - 16r + 3 = (5r - 1)(r - 3)$

But numerator $3r^2 - 4r - 6$ does NOT factor with integer coefficients, and no common factors with denominator.

So this expression cannot be simplified further.

✔ Final for #7: $\frac{3r^2 - 4r - 6}{5r^2 - 16r + 3}$ (already simplified)

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8. $\frac{10k^2 - 29k + 21}{2k^2 - 7k + 6}$

Factor numerator:
$10k^2 - 29k + 21$ → Multiply 10·21=210, find factors of 210 that add to -29 → -14 and -15
So: $10k^2 - 14k - 15k + 21 = 2k(5k - 7) -3(5k - 7) = (2k - 3)(5k - 7)$

Factor denominator:
$2k^2 - 7k + 6$ → Multiply 2·6=12, factors of 12 that add to -7 → -3 and -4
So: $2k^2 - 4k - 3k + 6 = 2k(k - 2) -3(k - 2) = (2k - 3)(k - 2)$

Now:
$\frac{(2k - 3)(5k - 7)}{(2k - 3)(k - 2)} = \frac{\cancel{(2k - 3)}(5k - 7)}{\cancel{(2k - 3)}(k - 2)} = \frac{5k - 7}{k - 2}$

✔ Final for #8: $\frac{5k - 7}{k - 2}$

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Final Answer:
1. $\frac{n + 1}{n - 7}$
2. $h + 6$
3. $\frac{s - 9}{s - 3}$
4. $\frac{1}{g + 7}$
5. $c - 5$
6. $\frac{8h - 1}{4h + 3}$
7. $\frac{3r^2 - 4r - 6}{5r^2 - 16r + 3}$
8. $\frac{5k - 7}{k - 2}$