TL;DR: A rational algebraic expression is one polynomial divided by another. Every operation on them runs on the same three-step engine: factor everything, list the values that make the original denominator zero, then cancel or combine. You may cancel only whole factors that are multiplied, never terms that are added. When you solve a rational equation, multiplying out can create answers that are not really answers, so every root has to be checked against the restrictions before you write it down. The six worked examples below cover simplifying, multiplying, dividing, adding with unlike denominators, cross-multiplication, and a rate problem.
Most people searching for rational algebraic expression examples want the same thing: the actual steps, written out, on problems that look like the ones in their module. This guide gives you six, each worked line by line with a short note on why the step is there. The topic sits in Grade 8 Mathematics, first quarter, and it is the foundation for rational equations, rational functions and later work in Pre-Calculus, so the habits you build here follow you for years.
One rule is worth stating before anything else, because almost every mark lost in this topic traces back to it. Restrictions come from the original denominator, not the simplified one. Write them down before you cancel.
What Is a Rational Algebraic Expression?
A rational algebraic expression is a ratio of two polynomials, with the denominator not equal to the zero polynomial. The word "rational" comes from ratio, not from anything to do with reasonableness.
| Expression | Rational? | Why |
|---|---|---|
| (x + 2)/(x - 5) | Yes | Polynomial over polynomial |
| (x² - 9)/(x² + 7x + 12) | Yes | Polynomial over polynomial |
| 7/x | Yes | 7 is a constant polynomial |
| x² + 3x - 4 | Yes | Any polynomial is itself over 1 |
| (√x + 1)/(x - 2) | No | A variable under a radical is not a polynomial |
| (2^x)/(x + 1) | No | A variable in an exponent is not a polynomial |
The denominator restriction is not a technicality. The expression (x - 3)/(x + 4) is a perfectly good number for every value of x except -4, where it asks you to divide by zero and stops meaning anything. That single excluded value has to travel with the expression through every later step.
Where This Topic Sits in the DepEd Curriculum
Rational algebraic expressions are Grade 8 Mathematics, first quarter, in the Number and Algebra domain. Grade 8 moved onto the MATATAG K to 10 curriculum in SY 2025-2026: DepEd Order No. 010, s. 2024 states that the curriculum is implemented by phase, "starting with Kindergarten, Grades 1, 4, and 7 in School Year (SY) 2024-2025," followed by "Grades 2, 3, 5, and 8 in SY 2025-2026; Grades 6, 9, and 10 in SY 2026-2027."
Copies of the MATATAG Grade 8 Mathematics curriculum guide in circulation place the topic in Quarter 1 under Number and Algebra, with a content standard covering rational algebraic expressions and equations and competencies to simplify them, perform operations on them, and solve problems involving simple rational algebraic equations using cross-multiplication. We flag that as reported rather than quoted: DepEd's public MATATAG download page currently hosts the Phase 1 guides for Grades 1, 4 and 7, and the Grade 8 guide reaches schools through regional and division channels. Ask your teacher for the copy your school is actually using before treating any competency list as final.
In the outgoing 2016 K to 12 Mathematics Curriculum Guide, the same content carried explicit competency codes that still appear on many worksheets and learning materials in circulation:
| Code | Competency |
|---|---|
| M8AL-Ic-1 | Illustrates rational algebraic expressions |
| M8AL-Ic-2 | Simplifies rational algebraic expressions |
| M8AL-Ic-d-1 | Performs operations on rational algebraic expressions |
| M8AL-Id-2 | Solves problems involving rational algebraic expressions |
If your module carries M8AL codes, it was built for the older curriculum guide. The mathematics is unchanged, but the sequencing and the assessment wording may not match what your teacher is using now. For the wider picture of how the mathematics curriculum is organised, see our DepEd math curriculum guide explainer, the MATATAG curriculum guide for parents, and the K to 12 curriculum overview.
Worked Example 1: Simplifying
Simplify (x² - 9)/(x² + 7x + 12).
Step 1. Factor both parts completely. The numerator is a difference of two squares. The denominator is a general trinomial whose factors multiply to 12 and add to 7, so 3 and 4.
(x² - 9)/(x² + 7x + 12) = [(x - 3)(x + 3)] / [(x + 3)(x + 4)]
Step 2. List the restrictions from the original denominator. Set each denominator factor to zero: x + 3 = 0 gives x = -3, and x + 4 = 0 gives x = -4. So x ≠ -3 and x ≠ -4.
Step 3. Divide out the common factor. The factor (x + 3) appears on top and bottom, and it is a factor, not a term, so it may be cancelled.
Answer: (x - 3)/(x + 4), where x ≠ -3 and x ≠ -4
Notice that -4 is visible in the final answer but -3 is not. It is still excluded. This is exactly why step 2 happens before step 3.
Worked Example 2: Multiplying
Multiply (x² - 4)/(x² + 6x + 9) × (x + 3)/(x - 2).
Factor first, always. The numerator of the first fraction is a difference of squares, and its denominator is a perfect square trinomial.
[(x - 2)(x + 2)] / [(x + 3)(x + 3)] × (x + 3)/(x - 2)
Restrictions come from every denominator that appears in the problem: x ≠ -3 and x ≠ 2.
Cancel one (x + 3) and one (x - 2) across the product:
Answer: (x + 2)/(x + 3), where x ≠ -3 and x ≠ 2
The reason you may cancel diagonally here, and not in an addition problem, is that multiplication of fractions puts everything into a single numerator and a single denominator. Nothing is being added, so every factor is fair game.
Worked Example 3: Dividing
Divide (x² - 25)/(3x) ÷ (x + 5)/(9x²).
Division is multiplication by the reciprocal. Flip the second fraction, then proceed exactly as in Example 2.
(x² - 25)/(3x) × 9x²/(x + 5) = [(x - 5)(x + 5) × 9x²] / [3x(x + 5)]
Restrictions: 3x ≠ 0 gives x ≠ 0; the divisor's denominator 9x² ≠ 0 gives x ≠ 0 again; and because you cannot divide by zero, the divisor itself cannot be zero, so x + 5 ≠ 0 gives x ≠ -5. Cancel (x + 5), and reduce 9x²/3x to 3x.
Answer: 3x(x - 5), or 3x² - 15x, where x ≠ 0 and x ≠ -5
Students routinely lose the x ≠ -5 restriction here because (x + 5) is in the divisor's numerator, not in a denominator. It still has to be excluded, because dividing by zero is what the original problem would be asking you to do.
Worked Example 4: Adding With Unlike Denominators
Add 3/(x + 2) + 5/(x - 1).
The denominators share no factors, so the least common denominator is simply their product, (x + 2)(x - 1). Restrictions: x ≠ -2 and x ≠ 1.
Rewrite each fraction over the LCD by multiplying its numerator by whatever factor it is missing:
[3(x - 1)] / [(x + 2)(x - 1)] + [5(x + 2)] / [(x + 2)(x - 1)]
Now add the numerators only, and keep the LCD:
(3x - 3 + 5x + 10) / [(x + 2)(x - 1)] = (8x + 7) / [(x + 2)(x - 1)]
Answer: (8x + 7)/[(x + 2)(x - 1)], where x ≠ -2 and x ≠ 1
Do not "simplify" this by cancelling anything. The numerator 8x + 7 is a sum, so it has no factor in common with the denominator, and nothing cancels.
A subtraction variant. For 2/(x² - 4) - 1/(x + 2), factor first: x² - 4 = (x - 2)(x + 2), so the LCD is (x - 2)(x + 2). The second fraction needs (x - 2):
[2 - (x - 2)] / [(x - 2)(x + 2)] = (4 - x)/[(x - 2)(x + 2)], where x ≠ 2 and x ≠ -2
The brackets around (x - 2) matter. Forgetting to distribute the minus sign across both terms is the most common arithmetic slip in subtraction problems.
Worked Example 5: Solving by Cross-Multiplication
Solve 3/(x - 2) = 5/(x + 4).
One fraction equals one fraction, so cross-multiply. Restrictions first: x ≠ 2 and x ≠ -4.
3(x + 4) = 5(x - 2)
3x + 12 = 5x - 10
22 = 2x
x = 11
Check it against the restrictions: 11 is neither 2 nor -4, so it is a genuine solution. Substituting back, 3/9 = 1/3 and 5/15 = 1/3, which agree.
Now the case that catches people. Solve x/(x - 3) = 3/(x - 3), with restriction x ≠ 3.
x(x - 3) = 3(x - 3)
x² - 3x = 3x - 9
x² - 6x + 9 = 0
(x - 3)² = 0
x = 3
The algebra is correct at every line, and the answer is still wrong. x = 3 is excluded, so it is an extraneous root and the equation has no solution. This is why "check against the restrictions" is a required final step, not an optional flourish.
Worked Example 6: A Rate Problem
This is an illustrative problem constructed for this guide, not data from a real school or project. It goes a step past cross-multiplication, so treat it as a stretch item.
One pump fills a tank in x hours. A slower pump takes 3 hours longer. Working together they fill it in 2 hours. How long does each take alone?
Work-rate problems become rational equations because a rate is "one job per x hours," which is 1/x.
1/x + 1/(x + 3) = 1/2, with x ≠ 0 and x ≠ -3
Multiply every term by the LCD, 2x(x + 3):
2(x + 3) + 2x = x(x + 3)
2x + 6 + 2x = x² + 3x
4x + 6 = x² + 3x
x² - x - 6 = 0
(x - 3)(x + 2) = 0
x = 3 or x = -2
A pump cannot take -2 hours, so reject that root on physical grounds and keep x = 3. The faster pump takes 3 hours, the slower one 6. Check: 1/3 + 1/6 = 1/2. Correct.
Two different reasons to reject a root appear in this guide: an excluded value from the restrictions (Example 5) and an answer that makes no sense in context (here). Both count.
The Whole Topic in One Table
| Operation | First move | What you may cancel | Watch for |
|---|---|---|---|
| Simplify | Factor top and bottom | Common factors of the whole numerator and denominator | Restrictions from the original denominator |
| Multiply | Factor everything, then cancel across | Any factor on top against any factor on the bottom | Restrictions from every denominator in the problem |
| Divide | Multiply by the reciprocal | Same as multiplication | The divisor's numerator also becomes a restriction |
| Add or subtract | Factor denominators, build the LCD | Nothing, until after the numerators are combined | Distributing the minus sign across the whole numerator |
| Solve an equation | Cross-multiply or clear the LCD | Not applicable, you are solving, not simplifying | Extraneous roots that violate the restrictions |
Common Mistakes That Cost Marks
- Cancelling terms instead of factors. In (x + 3)/x the x on the bottom is a factor of the denominator, but on top it is only a term inside a sum. Nothing cancels. Test yourself with numbers: (2 + 3)/2 is 2.5, not 3.
- Dropping restrictions after simplifying. If (x + 3) cancels, x ≠ -3 still holds. Markers look specifically for this.
- Forgetting the brackets in subtraction. 2/(x² - 4) - 1/(x + 2) becomes [2 - (x - 2)], not [2 - x - 2].
- Using the product of the denominators as the LCD when they share factors. For 1/(x² - 4) + 1/(x + 2) the LCD is (x - 2)(x + 2), not (x² - 4)(x + 2). The answer is not wrong, but it is not in lowest terms and it usually costs a mark.
- Cross-multiplying a sum. Cross-multiplication works only when a single fraction equals a single fraction. If either side has two terms, clear the LCD instead.
- Skipping the final check on an equation. Every root has to be tested against the restrictions before it counts as a solution.
How to Practise This Properly
- Factor before you do anything else. Almost every step in this topic is unlocked by factoring, and almost every failure in it starts with skipping that step.
- Write the restrictions on the same line as the original problem. Not at the end. If they are recorded first, you cannot lose them.
- Substitute a number to test a suspicious cancellation. Pick x = 1 or x = 2, evaluate the original and your simplified version, and see whether they agree.
- Practise subtraction more than addition. The sign errors live there.
- Do a handful of extraneous-root problems on purpose, so that "no solution" stops feeling like a mistake and starts feeling like an answer.
If you are heading into a course that uses this constantly, mathematics, engineering, accountancy or the sciences, our STEM guide covers what that pathway looks like in senior high school. One caveat on terminology: DepEd Order No. 017, s. 2026 replaces the named strands with elective clusters for Grade 11 from SY 2026-2027, so the science and mathematics path is now a cluster within the Academic Track rather than a strand. Our guide to the Strengthened SHS Curriculum explains what changed and when.
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Disclaimer: curriculum guides and competency lists are revised by DepEd from time to time. Confirm which curriculum version your class is following with your teacher or school before relying on any code list, including this one.
Sources
- DepEd — DO 010, s. 2024, Policy Guidelines on the Implementation of the MATATAG Curriculum — phased implementation by grade level and school year, quoted above from Section II, Scope
- DepEd — DO 010, s. 2024 (full PDF)
- DepEd — K to 12 Mathematics Curriculum Guide, August 2016 — Grade 8 competency codes M8AL-Ic-1, M8AL-Ic-2, M8AL-Ic-d-1, M8AL-Id-2
- DepEd — MATATAG Curriculum resources hub
- SchoolFinderPH — DepEd Math Curriculum Guide
- SchoolFinderPH — What Is the MATATAG Curriculum?
- SchoolFinderPH — K to 12 Curriculum Guide
