Going Deeper with Addition and Subtraction of Polynomials
In Professor Baker's Math Class, we're not just about memorizing rules; we're about understanding why things work the way they do. This lesson dives into the addition and subtraction of polynomials, exploring different methods and the reasoning behind them.
To add or subtract polynomials, the key is to combine like terms. Like terms have the same variable raised to the same power. You can arrange your work either vertically or horizontally; the choice is yours!
Adding Polynomials
Let's consider an example:
Add $ (4x^2 + 12x + 6) + (5x^2 - 6x + 9) $
Vertical Arrangement:
Write the polynomials, aligning like terms:
$$ \begin{array}{cccc} 4x^2 & +12x & +6 \\ 5x^2 & -6x & +9 \\ \hline \end{array} $$
Add the coefficients of like terms:
$$ \begin{array}{cccc} 4x^2 & +12x & +6 \\ 5x^2 & -6x & +9 \\ \hline 9x^2 & +6x & +15 \end{array} $$
Horizontal Arrangement:
Write the polynomials:
$(-7x^2 + 2x) + (x^2 - 2x + 5)$
Group like terms:
$(-7x^2 + x^2) + (2x - 2x) + 5$
Simplify:
$ -6x^2 + 0x + 5 = -6x^2 + 5$
Key takeaway: Whether you add vertically or horizontally, the result is the same! The method you choose depends on your personal preference and the complexity of the polynomials.
Subtracting Polynomials
Subtracting polynomials is similar to adding, but with an important twist: you add the opposite of the polynomial being subtracted.
Consider this example:
Subtract $ (2 + 9x^2) - (-6x^2 - 3x + 1) $
Vertical Arrangement:
Write the first polynomial in standard form, aligning like terms:
$$ \begin{array}{cccc} 9x^2 & & +2 \\ -(-6x^2 & -3x & +1) \\ \hline \end{array} $$
Add the opposite of the second polynomial:
$$ \begin{array}{cccc} 9x^2 & & +2 \\ 6x^2 & +3x & -1 \\ \hline \end{array} $$
Add the coefficients of like terms:
$$ \begin{array}{cccc} 9x^2 & & +2 \\ 6x^2 & +3x & -1 \\ \hline 15x^2 & +3x & +1 \end{array} $$
Horizontal Arrangement:
Write the polynomials:
$(6x^3 + 3x^2 + 2x + 9) - (4x^3 + 6x^2 - 2x + 7)$
Add the opposite:
$(6x^3 + 3x^2 + 2x + 9) + (-4x^3 - 6x^2 + 2x - 7)$
Group like terms:
$(6x^3 - 4x^3) + (3x^2 - 6x^2) + (2x + 2x) + (9 - 7)$
Simplify:
$2x^3 - 3x^2 + 4x + 2$
Remember to pay close attention to the signs when subtracting. A common mistake is forgetting to distribute the negative sign to all terms in the second polynomial.
Keep practicing, and you'll become a polynomial pro in no time! Good luck with your homework, and see you in the next class!