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Adding and subtracting polynomials with two variables review

Adding and subtracting polynomials is all about combining like terms. When the terms have two variables, it gets a little bit tricky to figure out which terms are like terms. In this article, we review some examples and give you a chance for you to practice the skill yourself.
Adding and subtracting polynomials is all about combining like terms. The polynomials in this article have two variables, which makes figuring out which terms are like terms a little more difficult.

Example 1

Simplify.
left parenthesis, minus, 7, d, cubed, minus, 5, d, cubed, f, cubed, minus, 3, f, cubed, right parenthesis, minus, left parenthesis, 8, d, cubed, plus, 4, d, cubed, f, cubed, plus, 2, f, cubed, right parenthesis
Rewrite without parentheses:
start color #11accd, minus, 7, d, cubed, minus, 5, d, cubed, f, cubed, minus, 3, f, cubed, end color #11accd, start color #ca337c, minus, 8, d, cubed, minus, 4, d, cubed, f, cubed, minus, 2, f, cubed, end color #ca337c
Group like terms:
left parenthesis, start color #11accd, minus, 7, d, cubed, end color #11accd, start color #ca337c, minus, 8, d, cubed, end color #ca337c, right parenthesis, plus, left parenthesis, start color #11accd, minus, 5, d, cubed, f, cubed, end color #11accd, start color #ca337c, minus, 4, d, cubed, f, cubed, end color #ca337c, right parenthesis, plus, left parenthesis, start color #11accd, minus, 3, f, cubed, end color #11accd, start color #ca337c, minus, 2, f, cubed, end color #ca337c, right parenthesis
Simplify:
minus, 15, d, cubed, minus, 9, d, cubed, f, cubed, minus, 5, f, cubed

Example 2

Simplify.
left parenthesis, u, squared, plus, 5, u, squared, v, plus, 4, u, v, squared, right parenthesis, minus, left parenthesis, u, squared, plus, 5, u, squared, v, squared, plus, 4, u, v, right parenthesis
Rewrite without parentheses:
start color #11accd, u, squared, plus, 5, u, squared, v, plus, 4, u, v, squared, end color #11accd, start color #ca337c, minus, u, squared, minus, 5, u, squared, v, squared, minus, 4, u, v, end color #ca337c
Group like terms:
left parenthesis, start color #11accd, u, squared, end color #11accd, start color #ca337c, minus, u, squared, end color #ca337c, right parenthesis, start color #11accd, plus, 5, u, squared, v, plus, 4, u, v, squared, end color #11accd, start color #ca337c, minus, 5, u, squared, v, squared, minus, 4, u, v, end color #ca337c
Simplify:
minus, 5, u, squared, v, squared, plus, 5, u, squared, v, plus, 4, u, v, squared, minus, 4, u, v
Want to see another example? Check out this video.

Practice

Problem 1
Simplify.
left parenthesis, minus, 2, x, start superscript, 4, end superscript, plus, 7, x, squared, y, minus, 7, right parenthesis, plus, left parenthesis, minus, 9, x, cubed, plus, 7, x, y, plus, 7, right parenthesis

Want more practice? Check out this exercise.

Want to join the conversation?

  • starky seed style avatar for user El Ouadi, Jordan
    how you do this
    (6 votes)
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  • spunky sam blue style avatar for user 💎Chυcκ Lørrε💎
    Why it's -15d³-9d³f³-5f³ instead of -24d³-4f³? -15d³ and -9d³ is like terms! -5f³ and f³ is also like terms! It could be simplify more!
    (5 votes)
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  • aqualine ultimate style avatar for user GIANNA TERITO
    how do i solve this without getting confused
    (2 votes)
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    • aqualine ultimate style avatar for user nr0193954
      I know it looks very confusing when simplifying these polynomials. However, what I would do is try to solve the equation like I would with any other expression dealing with polynomials. Most importantly, I would carefully analyze the expression mostly for the signs. This is because not doing so can mess up everything and therefore make you more confused.
      (1 vote)
  • aqualine ultimate style avatar for user Jacoblevin
    when you put them in order which would go first say

    4g^2h . or 4gh^2
    (1 vote)
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  • blobby green style avatar for user Carter Hooks
    what is the degree and leading coefficient of 2x^2 y - 7x^2 y^3 +6xy^2
    (1 vote)
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  • blobby green style avatar for user Savannah Pettry
    Where does the +7 go to when you add or subtract the parentheses?
    (0 votes)
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  • blobby green style avatar for user Marty Rogers
    Does this get better with practice? When subtracting, if the subtraction sign (or negative sign) is outside of a parenthesis, I distribute the negative sign to each of the terms inside the parenthesis, but sometimes mistake the sign. When rewriting a problem without parenthesis I mistake the degree, e.g. x^4 is correct but I may write x^2, and I am not aware of the mistake until I check my math.
    (0 votes)
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    • duskpin ultimate style avatar for user Quinn
      Yes, everything in math will get better with practice!

      Basically what he is doing is also known as the Distributive law. You can look it up online but basically, it is actually just multiplying every term in the bracket with a number (can be a negative or a positive number).

      Therefore, for example -(x^4+3x-6)
      can be written as -1 (x^4+3x-6), both are equivalent.

      And recall that when we multiply, the rules on whether the result will have a positive or negative sign when there is a negative number involved in the equation is that;
      1) when there is an even number of negative number being multiplied, the result will always be a positive number.
      2) if there is an odd number of negative number being multiplied, the result will always be a negative number.

      Mistaking the degree is part of human error or aka careless mistake, for mistakes like this, just check through your working again before submitting your answer
      (2 votes)
  • starky tree style avatar for user Brenna
    would -2x^4-9x^3+xy be the answer
    (0 votes)
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