Simplify Rational Expressions worksheet with eight algebraic fractions to reduce.

Let’s simplify each rational expression one by one. We’ll factor the numerators and denominators, 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:
$n^2 - n - 42 = (n - 7)(n + 6)$

Now write as:
$\frac{(n + 1)(n + 6)}{(n - 7)(n + 6)}$

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 $h + 7$

So:
$\frac{(h + 6)(h + 7)}{h + 7}$

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

Write as:
$\frac{(s - 9)(s + 5)}{(s + 5)(s - 3)}$

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

✔ 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 $g + 9$

So:
$\frac{g + 9}{(g + 7)(g + 9)}$

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 $c - 3$

So:
$\frac{(c - 3)(c - 5)}{c - 3}$

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? Wait — let's try factoring by grouping or trial.

Try:
$8h^2 + 15h - 2$

Look for factors of $8 \cdot (-2) = -16$ that add to 15 → 16 and -1.

Split middle term:
$8h^2 + 16h - h - 2$

Group:
$(8h^2 + 16h) + (-h - 2) = 8h(h + 2) -1(h + 2) = (8h - 1)(h + 2)$

Check: $(8h - 1)(h + 2) = 8h^2 + 16h - h - 2 = 8h^2 + 15h - 2$ ✔

Factor denominator:
$4h^2 + 11h + 6$

Find two numbers that multiply to $4 \cdot 6 = 24$, add to 11 → 8 and 3.

Split:
$4h^2 + 8h + 3h + 6 = 4h(h + 2) + 3(h + 2) = (4h + 3)(h + 2)$

Now write full fraction:
$\frac{(8h - 1)(h + 2)}{(4h + 3)(h + 2)}$

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: discriminant = $(-16)^2 - 4(5)(3) = 256 - 60 = 196 = 14^2$ → does factor.

But since numerator doesn’t factor with integer coefficients, and no common factors visible, this expression cannot be simplified further.

Wait — double-check numerator: maybe I made a mistake?

Try factoring $3r^2 - 4r - 6$:
Possible pairs: (3r ?)(r ?) → constants must multiply to -6.

Try: (3r + 2)(r - 3) = 3r² -9r +2r -6 = 3r² -7r -6 ✘
(3r - 2)(r + 3) = 3r² +9r -2r -6 = 3r² +7r -6 ✘
(3r + 3)(r - 2) = 3r² -6r +3r -6 = 3r² -3r -6 ✘
No integer factors. So yes, numerator doesn't factor.

Denominator: $5r^2 -16r +3$

Factors of 5*3=15 that add to -16 → -15 and -1

Split: $5r^2 -15r - r + 3 = 5r(r - 3) -1(r - 3) = (5r -1)(r - 3)$

So denominator = $(5r -1)(r - 3)$

Numerator: $3r^2 -4r -6$ — no common factors with denominator.

So this expression is already in simplest form.

✔ Final for #7: $\frac{3r^2 - 4r - 6}{5r^2 - 16r + 3}$ (cannot simplify)

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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 two numbers that multiply to 210, add to -29 → -14 and -15.

Split:
$10k^2 -14k -15k +21 = 2k(5k -7) -3(5k -7) = (2k -3)(5k -7)$

Check: $(2k -3)(5k -7) = 10k^2 -14k -15k +21 = 10k^2 -29k +21$ ✔

Factor denominator:
$2k^2 -7k +6$

Multiply 2*6=12. Numbers that multiply to 12, add to -7 → -3 and -4.

Split:
$2k^2 -3k -4k +6 = k(2k -3) -2(2k -3) = (k -2)(2k -3)$

Wait — check:
$k(2k -3) -2(2k -3) = (k -2)(2k -3)$ ✔

Now write full fraction:
$\frac{(2k -3)(5k -7)}{(k -2)(2k -3)}$

Cancel $(2k -3)$:
$\frac{5k -7}{k -2}$

✔ 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. $\frac{5k - 7}{k - 2}$