Addition and Subtraction of Polynomials
Algorithms for Addition and Subtraction of Polynomials
To add two polynomials, just put a “+” sign between them. Three or more polynomials are added in the same way.
To find the difference of two polynomials, enclose the second polynomial in parentheses, put a “-” sign in front of it, and expand the parentheses.
Three or more polynomials are subtracted similarly.
The sum (or, the difference) of polynomials is also a polynomial.
About the rules for expanding parentheses – see §4 of this Guide.
Notice!
Do not forget to change the signs when subtracting polynomials!
Wrong: (a+b)-(c+d) ≠ a+b-c+d
Right: (a+b)-(c+d) = a+b-c-d
To avoid mistakes, use parentheses when subtracting, and expand parentheses in accordance with the rules.
Examples of Problem Solving
Example 1. Find the sum and the difference of the polynomials:
a) $ 9x^3-7x^2+8 и 5x^2+x-2$
Sum:
$ 9x^3-7x^2+8+5x^2+x-2 = 9x^3+(-7+5) x^2+x+(8-2)=9x^3-2x^2+x+6 $
Difference:
$ 9x^3-7x^2+8-(5x^2+x-2) = 9x^3-7x^2+8-5x^2-x+2 = $
$ = 9x^3+(-7-5) x^2-x+(8+2) = 9x^3-12x^2-x+10 $
b) $-\frac{3}{5} xy+xy^2-\frac{1}{6} и 1 \frac{2}{3} xy-xy^2+1 $
Sum:
$-\frac{3}{5} xy+xy^2-\frac{1}{6}+ 1 \frac{2}{3} xy-xy^2+1 = (-\frac{3}{5}+1 \frac{2}{3})xy+0 \cdot xy^2+(- \frac{1}{6}+1)=$
$ =(1+\frac{10-9}{15})xy+\frac{5}{6} = 1 \frac{1}{15} xy+ \frac{5}{6} $
Difference:
$ -\frac{3}{5} xy+xy^2-\frac{1}{6}- (1 \frac{2}{3} xy-xy^2+1) = -\frac{3}{5} xy+xy^2-\frac{1}{6}- 1\frac{2}{3} xy+xy^2-1 = $
$ = (-\frac{3}{5}-1 \frac{2}{3})xy+(1+1)xy^2+(-\frac{1}{6} -1) = (-1-\frac{9+10}{15})xy+2xy^2-1 \frac{1}{6} = $
$ =-2 \frac{4}{15} xy+2xy^2-1 \frac{1}{6} $
Example 2. Evaluate the expression for x = -5
a) $ 16x^4-x^3-(15x^4-x^3+1) = 16x^4-x^3-15x^4+x^3-1 = x^4-1 $
Substitute: $ (-5)^4-1 = 625-1 = 624 $
b) $ 3x^3-2x^2+x-8-(x^3-2x^2+x) = 3x^3-2x^2+x-8-x^3+2x^2-x = 2x^3-8 $
Substitute: $ 2 \cdot (-5)^3-8 = 2 \cdot (-125)-8 = -258 $
Example 3. Find the value of x such that the numerical value of the trinomial $ 2x^2+5x-3$ is greater than the numerical value of the binomial $2x^2+1$ by 4?
As stated: $ (2x^2+5x-3 )-(2x^2+1) = 4 $
$ 2x^2+5x-3-2x^2-1 = 4 $
5x-4 = 4
5x = 8
$ x = \frac{8}{5} = 1,6 $
Answer: 1,6
Example 4. What polynomial must be added to the polynomial $ 2x^2+x-16 $ so that the resulting polynomial is $x^3-x^2+2$?
Denote the unknown polynomial by P(x). As stated:
$ 2x^2+x-16+P(x) = x^3-x^2+2 $
$ P(x) = x^3-x^2+2-(2x^2+x-16) = x^3-x^2+2-2x^2-x+16 = x^3-3x^2-x+18 $
Answer: $ x^3-3x^2-x+18 $
