8.1: Polynomial Addition and Subtraction (and Combining Like Terms)
To add and subtract polynomials, combine like terms . Like terms have the same variables with the same exponents. The coefficients of the terms may be different.
Be careful when subtracting, to distribute the subtraction (think of it as an addition of \((−1)\) times the polynomial).
Add or Subtract the Polynomials:
- \((−6a^3 + 5a^2 − 7a − 9) + (3a^3 + 5a^2 + a + 8)\)
- \((4x^2 − 3) + (3x^2 − 8x + 7)\)
- \((3x^2 − 4x + 6) − (2x^2 − x − 9)\)
- \((−4x^3 + 5x^2 + 15) − (2x^2 − 4x + 9)\)
Solution
- \(\begin{array} &&(−6a^3+5a^2−7a−9)+(3a^3+5a^2+a+8) &\text{Example problem} \\ &−6a^3 + 3a^3 + 5a^2 + 5a^2 − 7a + a − 9 + 8 &\text{Pair like terms together} \\ &−3a^3 + 10a 2 − 6a − 1 &\text{Solution} \end{array}\)
- \(\begin{array} &&(4x^2 − 3) + (3x^2 − 8x + 7) &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Example problem} \\ &4x^2 + 3x^2 − 8x − 3 + 7 &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Pair like terms together} \\ &7x^2 − 8x + 4 &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Solution} \end{array}\)
- \(\begin{array} &&(3x^2 − 4x + 6) − (2x^2 − x − 9) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&\text{Example problem} \\ &3x^2 − 2x^2 − 4x + x + 6 + 9 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&\text{Distribute the subtraction and pair like terms together} \\ &x^2 − 3x + 15 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&\text{Solution} \end{array}\)
- \(\begin{array} &&(−4x^3 + 5x^2 + 15) − (2x^2 − 4x + 9) \;\;\;\;\;\;\;\;\;\;\;&\text{Example problem} \\ &−4x^3 + 5x^2 − 2x^2 + 4x + 15 − 9 \;\;\;\;\;\;\;\;\;\;\;&\text{Distribute the subtraction and pair like terms together} \\ &−4x^3 + 3x^2 + 4x + 6 \;\;\;\;\;\;\;\;\;\;\;&\text{Solution} \end{array}\)
Add or Subtract the Polynomials
- \((5x^2 + 8) − (x^2 + 4x + 3)\)
- \((x^3 − 14x^2 ) − (−4x^3 + 5x^2 + 8)\)
- \((6x^2 + 7x − 9) + (−9x^2 + 2)\)
- \((x^3 + 6x^2 − 8x + 14) + (9x^3 − 7x^2 + 5x − 11)\)
- \((3x − 4) − (2x^2 − 3x − 9)\)
- \((2x^2 + x + 3) − (5x^2 − 1)\)