Simplify Rational Expressions Worksheets (printable, online ... - Free Printable

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

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

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

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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 $(h + 7)$:
→ $h + 6$

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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 $(s + 5)$:
→ $\frac{s - 9}{s - 3}$

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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 $(g + 9)$:
→ $\frac{1}{g + 7}$

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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 $(c - 3)$:
→ $c - 5$

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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
Split middle term:
$8h^2 + 16h - h - 2 = 8h(h + 2) -1(h + 2) = (8h - 1)(h + 2)$

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

Cancel $(h + 2)$:
→ $\frac{8h - 1}{4h + 3}$

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

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

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

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

So factors: $(r - 3)(5r - 1)$? Let’s check:
$(r - 3)(5r - 1) = 5r^2 - r - 15r + 3 = 5r^2 - 16r + 3$ ✔

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

So this expression cannot be simplified further.

Wait — let me double-check if I made a mistake.

Try factoring numerator again:
$3r^2 - 4r - 6$
Possible pairs: (3, -2), (1, -6), etc. None give -4 when cross-multiplied.

Yes, it doesn’t factor nicely. So we leave it as is.

BUT — maybe I should check if there's a typo? The problem says “simplify”, so perhaps it was meant to factor? Let me try grouping or other methods.

Alternatively, maybe use quadratic formula for numerator:

Roots: $r = \frac{4 \pm \sqrt{16 + 72}}{6} = \frac{4 \pm \sqrt{88}}{6} = \frac{4 \pm 2\sqrt{22}}{6} = \frac{2 \pm \sqrt{22}}{3}$ — irrational, so no nice factors.

So yes, this fraction cannot be simplified.

But wait — looking back at the original image, maybe I misread? Let me recheck the problem.

Original: $\frac{3r^2 - 4r - 6}{5r^2 - 16r + 3}$

Hmm. Maybe it’s supposed to be $3r^2 - 4r - 4$ or something else? But as written, it doesn’t simplify.

However, let me check if denominator factors correctly:
$5r^2 - 16r + 3 = (5r - 1)(r - 3)$ — yes.

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

So answer is itself.

But let’s hold on — maybe I made a mistake in assuming it can't be simplified. Perhaps there’s a common factor I missed?

Let me plug in a value to test.

Let r = 0: numerator = -6, denominator = 3 → ratio = -2

If simplified form existed, it should match.

But since no common factors, we leave it.

Actually — wait! Let me try factoring numerator again using AC method.

A=3, C=-6 → AC = -18

Find two numbers that multiply to -18, add to -4 → -6 and 2

So: $3r^2 - 6r + 2r - 6 = 3r(r - 2) + 2(r - 2) = (3r + 2)(r - 2)$

Oh! I missed that earlier!

Let me verify:
$(3r + 2)(r - 2) = 3r^2 - 6r + 2r - 4 = 3r^2 - 4r - 4$ — but we have -6, not -4. Mistake!

Wait, original is $3r^2 - 4r - 6$

AC = 3 * (-6) = -18

Numbers that multiply to -18, add to -4: let's list:

-6 and 3 → sum -3
-9 and 2 → sum -7
-18 and 1 → sum -17
6 and -3 → sum 3
No pair adds to -4.

So indeed, doesn’t factor over integers.

Thus, this expression is already simplified.

But let me check online or think differently — perhaps the problem has a typo? In many textbooks, such problems are designed to simplify.

Looking at denominator: $5r^2 - 16r + 3 = (5r - 1)(r - 3)$

Suppose numerator was $3r^2 - 10r + 3$ or something — but it’s not.

Perhaps it’s $3r^2 - 4r - 4$? Then it would be (3r+2)(r-2), still no common factor.

Or $3r^2 - 11r + 6$? Not matching.

I think we have to accept that this one doesn’t simplify.

But let me move on and come back.

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

Factor numerator:
$10k^2 - 29k + 21$

AC = 10*21 = 210

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

Because (-14)*(-15)=210, (-14)+(-15)=-29

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

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

AC = 12, find numbers multiply to 12, add to -7 → -3 and -4

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

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

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

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Now back to #7: $\frac{3r^2 - 4r - 6}{5r^2 - 16r + 3}$

After careful checking, numerator does not factor over integers, and no common factors with denominator. So it remains as is.

But let me confirm with another approach.

Compute GCD of numerator and denominator polynomials.

Use polynomial division or Euclidean algorithm.

Let f(r) = 3r² - 4r - 6
g(r) = 5r² - 16r + 3

Divide f by g? Degree same, so divide leading coefficients.

Or compute resultant or just evaluate at points.

At r=1: num=3-4-6=-7, den=5-16+3=-8 → no common root

r=2: num=12-8-6=-2, den=20-32+3=-9

r=3: num=27-12-6=9, den=45-48+3=0 → undefined, but denominator zero at r=3, numerator at r=3 is 9≠0, so no common root.

r=1/5: den=5*(1/25)-16*(1/5)+3=1/5 - 16/5 + 15/5=0, num=3*(1/25)-4*(1/5)-6=3/25 - 20/25 - 150/25= -167/25 ≠0

So no common roots, hence no common factors.

Therefore, expression is already simplified.

But in the context of a worksheet titled "Simplify Rational Expressions", it's unusual to have one that doesn't simplify. Perhaps I misread the problem.

Looking back at the user's image description — the seventh expression is:

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

Maybe it's $3r^2 - 4r - 4$? Or $3r^2 - 10r - 8$? But as given, it's -6.

Another possibility: perhaps it's $3r^2 - 4r + 1$ or something. But I have to go with what's written.

Perhaps the constant is -4 instead of -6? Let me assume for a second it's -4.

If numerator were $3r^2 - 4r - 4$, then as before, (3r+2)(r-2)? 3r*r=3r², 3r*(-2)=-6r, 2*r=2r, 2*(-2)=-4 → 3r² -4r -4 yes.

Denominator (5r-1)(r-3)

No common factors.

If numerator were $3r^2 - 10r + 3$, then (3r-1)(r-3), and denominator (5r-1)(r-3), so cancel (r-3) → (3r-1)/(5r-1)

That would make sense.

But in the image, it's clearly "-6".

Perhaps it's a typo in the problem, but since I have to solve as given, I'll state it doesn't simplify.

But let me check online or standard problems.

Upon second thought, let's try to factor $3r^2 - 4r - 6$ using quadratic formula:

r = [4 ± √(16 + 72)] / 6 = [4 ± √88]/6 = [4 ± 2√22]/6 = [2 ± √22]/3

Not rational, so no rational factors.

Similarly, denominator has rational roots, but no overlap.

So final decision: this expression cannot be simplified.

But to match the pattern, perhaps the intended problem was different. However, based on what's provided, I'll proceed.

So for #7, answer is itself.

But let's write all answers.

Summary:

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. $\frac{3r^2 - 4r - 6}{5r^2 - 16r + 3}$ (no simplification)

8. $\frac{5k - 7}{k - 2}$

For #7, perhaps I should box it as is.

But let me see if the denominator can be factored differently or if I missed something.

Another idea: perhaps the numerator is 3r^2 - 4r - 6, and denominator 5r^2 - 16r + 3, and they have a common factor if I consider rational roots, but as calculated, no.

Perhaps use long division, but that won't help for simplifying rational expressions; we need common factors.

I think it's correct.

So final answers:

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. $\frac{5k - 7}{k - 2}$