Ninth Grade · Math · week 20

Polynomials: add, subtract, multiply

Classify polynomials by degree and number of terms, write them in standard form, and add, subtract, and multiply them.

A polynomial is a sum of terms built from whole-number powers of a variable: monomial (one term), binomial (two), trinomial (three), or just "polynomial" for more. Its degree is the highest exponent present, and standard form lists the terms from highest degree down to lowest - 4x³ - 2x² + 7 is already standard; 7 - 2x² + 4x³ is the same polynomial, sloppily ordered.

Adding and subtracting polynomials means combining LIKE terms only - same variable, same exponent - exactly like combining 3 apples and 2 apples but never apples with oranges. Multiplying is the distributive property scaled up: every term of the first polynomial multiplies every term of the second (FOIL is just this rule for two binomials specifically). Two special products are worth memorizing on sight because they appear constantly: (a+b)(a-b) = a² - b² (difference of squares) and (a+b)² = a² + 2ab + b² (perfect square trinomial) - note the easy-to-forget middle term.

📋 For the grown-up teacher
Teach it (10–15 min):
  • Have them physically underline like terms in matching colors before combining anything - the sorting step prevents most errors.
  • On subtraction, make them rewrite it as "add the opposite of every term" first - distributing the minus sign is where most mistakes hide.
  • Draw the FOIL arrows (First, Outer, Inner, Last) on the first several problems until the pattern is automatic, then let it fade into "distribute everything."
  • Have them verify (x+3)² is NOT x²+9 by actually multiplying it out - the missing middle term is the single most common polynomial error in algebra.
Talk about it:
  • What does "degree" mean, and how do you find it from a messy (non-standard-form) polynomial?
  • Why do you only combine terms that match in BOTH variable and exponent?
  • Why does (x+3)² have three terms instead of two?

Mastery looks like: They classify and order any polynomial, and add, subtract, or multiply two polynomials correctly, including both special products.

Common stumbles: Adding exponents when combining like terms (3x²+x² becoming 4x⁴ instead of 4x²); forgetting the middle term in a squared binomial.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
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