Let's consider the following two polynomials.
P(x) &= 5x^3 + 2x^2 - 7x + 1
Q(x) &= 3x^2 - 9x - 4
Next, we will compute P(x)+Q(x) and P(x)-Q(x) using the vertical and horizontal formats.
Vertical Method
To use this method, we first write the polynomials in standard form. Then we write Q(x) under P(x) in such a way that we align like terms — terms with the same powers — in columns.
5x^3 + 2x^2 - 7x + 1
3x^2 - 9x - 4
l
→ P(x)
→ Q(x)
To add the polynomials we combine like terms.
5x^3 + 2x^2 - 7x + 1
( +) 3x^2 - 9x-4
5x^3 + 5x^2 - 16x - 3_(P(x)+Q(x))
To subtract the polynomials, we add the additive inverse of Q(x), which is -3x^2 + 9x+4 and combine like terms.
5x^3 + 2x^2 - 7x + 1
( +) -3x^2 + 9x+4
5x^3 - x^2 + 2x + 5_(P(x)-Q(x))
In summary, we can write the following instructions.
To add polynomials in vertical format we write the polynomials in standard form, align like terms in columns, and combine like terms.
To subtract polynomials in vertical format we write the polynomials in standard form, align like terms in columns, and subtract by adding the additive inverse of the polynomial to be subtracted.
Horizontal Method
Using this method, to add polynomials we collect like terms and then we combine them.
