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Day 1: Adding & Subtracting Polynomials
To be able to add and subtract polynomials by combining like terms.
To add or subtract polynomials, we must group and combine like terms.
Like Terms Are...
Terms that have the same variable raised to the same power.
Example: \(3x^2\) and \(-8x^2\) are like terms. \(5x\) and \(5x^3\) are NOT.
Let's find the sum of \((3x^2 + 5x - 4) + (2x^2 - 7x + 1)\).
Step 1: Group like terms together (or line them up vertically).
\((3x^2 + 2x^2)\) \(+ (5x - 7x)\) \(+ (-4 + 1)\)
Step 2: Combine the coefficients of the like terms.
\(5x^2 - 2x - 3\)
When you see a minus sign between parentheses, you must change the problem before you start!
We use a strategy called "Keep, Change, Flip".
This turns a tricky subtraction problem into an easy addition problem.
\((8x^2 - 2x + 9) - (5x^2 + 3x - 1)\)
The first polynomial stays exactly the same.
\(8x^2 - 2x + 9\)
The subtraction sign in the middle becomes addition.
→
+
Flip the sign of EVERY term in the second polynomial.
\(-5x^2 - 3x + 1\)
Let's find the difference of \((8x^2 - 2x + 9) - (5x^2 + 3x - 1)\).
Step 1: Rewrite the problem using Keep, Change, Flip.
\((8x^2 - 2x + 9) \boldsymbol{\textcolor{green}{+}} (-5x^2 - 3x + 1)\)
Step 2: Now it's an addition problem! Group and combine like terms.
\((8x^2 - 5x^2) + (-2x - 3x) + (9 + 1)\)
\(3x^2 - 5x + 10\)
Let's simplify these expressions as a class.
1. (Addition) \((4a^2 + 2a - 1) + (a^2 - 5a + 9)\)
2. (Subtraction) \((7y + 4) - (2y + 2)\)
Find the difference. Remember to Keep, Change, Flip first!
\((9k^2 + 2k) - (k^2 - k + 12)\)
\(8k^2 + 3k - 12\)
Try these problems on your handout, starting with green.
Green Level
ADD: \((x^2 + 2x + 1) + (3x^2 + 5x + 4)\)
Yellow Level
SUBTRACT: \((5y^2 - 3y + 2) - (2y^2 + y + 1)\)
Red Level
Subtract \((x-3)\) from the sum of \((4x+2)\) and \((2x+5)\).
On your handout or a piece of paper, simplify the following expression: