L3 · Adding and Subtracting Monomials (Like Terms)

Learning goal

“I can add and subtract monomials by combining like terms - operating on the coefficients only, leaving the variable part unchanged - and I can prove a simplification by evaluating both versions at the same value of x.”

Like terms — terms with the same variable and the same degree (review from L2): the only terms that can be combined.

Combine — to add or subtract like monomials into a single monomial by adding or subtracting their coefficients.

Additive inverse — the opposite of a term: the additive inverse of 5x³ is -5x³, and a term plus its inverse is 0. Subtracting a monomial means adding its additive inverse.

Worked example

Simplify 5x² + 3x². Step 1: Both terms are counts of the same unit, x². Five x²-units plus three more x²-units. Step 2: Add the counts (the coefficients): 5 + 3 = 8. Step 3: The unit itself does not change: the result is 8x². Step 4: Sanity-check with a number, x = 2: original 5(4) + 3(4) = 20 + 12 = 32; simplified 8(4) = 32 ✓. Combining never changes the value - it only shortens the writing. Say it plainly: adding like monomials is counting. You would not say 5 apples + 3 apples = 8 apple² - the unit stays an apple. The variable part is the unit; the coefficient is the count.

Check yourself — explain it

Why does the exponent stay 2 in 5x² + 3x² = 8x²?

Paired problem — example still visible

Simplify 9x³ - 4x³ -> 5x³ (coefficients 9 - 4 = 5; check at x = 2: 9(8) - 4(8) = 72 - 32 = 40 and 5(8) = 40 ✓)

Faded completion — last step is yours

"4x + 7x: coefficients 4 + 7 = 11, so the sum is ___" -> 11x

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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