← Algebra II

Unit 1: Polynomial Arithmetic and Division

Treating polynomials as numbers you can add, multiply, and divide, and reading what a leftover remainder actually means.

7 topics · 26 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Adding and Subtracting Polynomials

  1. How it worksHow do like terms line up when you add two polynomials together?
  2. Worked exampleFind the sum of (3x3 - 5x + 7) and (2x3 + 4x2 - x - 1).
  3. Common mistakeWhy can't you combine 3x2 and 3x into a single term?
  4. Going furtherWhat changes when you subtract one polynomial from another instead of adding?
Your turn1 of 2
Find the sum of (7x2 - 6x + 8) and (4x2 + 2x - 3).

Multiplying Polynomials

  1. How it worksHow does the distributive property expand into every term multiplying every other term?
  2. Worked exampleFind the product of (x + 3) and (x2 - 2x + 5).
  3. Common mistakeWhy doesn't (x + y)2 come out equal to x2 + y2?
  4. Going furtherWhat happens to the number of terms when you multiply three binomials together instead of two?
Your turn1 of 2
Find the product of (x + 7) and (x2 - 6x + 8).

Special Product Patterns

  1. How it worksHow does squaring a binomial turn into three terms instead of two?
  2. Worked exampleFind the product of (2x - 5)(2x + 5) using the difference of squares pattern.
  3. Common mistakeWhy isn't (a - b)2 the same as a2 - b2?
  4. Going furtherWhat happens to the difference-of-squares pattern when a and b are themselves polynomials?
Your turn1 of 2
Find the product of (7x - 7) and (7x + 7).

Dividing a Polynomial by a Monomial

  1. How it worksHow do you split a polynomial over a monomial denominator term by term?
  2. Worked exampleFind the quotient of (12x4 - 8x3 + 20x2) divided by 4x2.
  3. Worked exampleFind the quotient of (18x5 - 12x3 + 6x) divided by 6x.
Your turn1 of 2
Find the quotient of (15x4 - 11x3 + 17x2) divided by 3x2.

Polynomial Long Division

  1. How it worksWalk me through dividing a polynomial by a binomial using long division, step by step.
  2. Worked exampleFind the quotient and remainder when x3 - 2x2 - 5x + 6 is divided by x - 3.
  3. Common mistakeWhy doesn't a leftover remainder mean you did the division wrong?
  4. Going furtherWhat happens to the long division process when the dividend is missing a term, like x3 + 1?
Your turn1 of 2
Divide x3 + 4x2 - 6x + 8 by x - 3 using long division. What is the remainder?

Synthetic Division

  1. How it worksHow do the coefficients slide down and multiply their way through synthetic division?
  2. Worked exampleFind the quotient of x3 + 4x2 - 3x - 2 divided by x - 1 using synthetic division.
  3. Common mistakeWhy do you flip the sign of the number you divide by before starting synthetic division?
  4. Going furtherWhat breaks in synthetic division when you try to divide by something like x2 + 1 instead of x - a?
Your turn1 of 2
Use synthetic division to find the quotient when x3 + 4x2 - 6x - 8 is divided by x - 3, as a function of x.

The Remainder Theorem and Factor Theorem

  1. How it worksHow does evaluating a polynomial at x = a give the same answer as dividing by x - a?
  2. Worked exampleFind the remainder when x4 - 3x2 + 2 is divided by x + 2 using the remainder theorem.
  3. Ties togetherHow does the factor theorem connect polynomial division to finding a polynomial's zeros?
Your turn1 of 2
Use the remainder theorem to find the remainder when x4 - 7x2 + 6 is divided by x + 5.