Add, subtract and Simplify Polynomials

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We add and subtract polynomials by grouping like terms. We therefore first define like terms, how to group them and then how to add and subtract them in order to simplify polynomials. Examples and questions and their solutions are included.
An online calculator to expand ans simpliy polynomials may be used to check answers to the examples and exericses presented.

Like Terms in a Polynomial

Like terms in a polynomial are terms with the same variable(s) and the same power.

Example 1

a) \( \; 2 x \; \) and \( \; -6 x \; \) are like terms because they have the same variable \( \; x \; \) to the same power 1.
b) \( \; 2 x^2 \; \) and \( \; 3 x \; \) are NOT like terms because the variable \( \; x \; \) has power \( \; 2 \; \) in the first expression and power \( \; 1 \; \) in the second expression.
c) \( \; - x^3 \; \) and \( \; 3 x^3 \; \) are like terms because the same variable \( \; x \; \) has same power \( \; 3 \; \) in both expressions.
d) \( \; - y x^2 \; \) and \( \; 3 x^2 \; \) are NOT like terms because the first expression has two variables \( \; x \; \) and \( \; y \; \) and the second expression has one variable \( \; x \; \).
e) \( \; - y^2 x^4 \; \) and \( \; 3 x^4 y^2 \; \) are like terms because both expressions have same variables \( \; x \; \) and \( \; y \; \) to the same powers \( \; 4 \; \) and \( \; 2 \; \)respectively.

Group Like Terms in a Polynomial

Example 2

Add and simplify \( \; 4 x + 6 x \; \)
Solution
Factor \( \; x \; \) out
\( \; 4 x + 6 x = x ( 4 + 6) \; \)
and add \( \; 4 \; \) and \( \; 6 \; \)
\( \; 4 x + 6 x = 10 x \; \)

Example 3

Add and simplify \( \; - x^3 + 9 x^3 \; \)
Solution
Factor \( \; x ^3 \; \) out and add coefficients
\( \; - x^3 + 9 x^3 = x^3 ( -1 + 9) \; \)
\( \; = 8 x^3 \; \)

Example 4

Add and simplify \( \; 3 y^2 x^4 - 4 x^4 y^2 \; \)
Solution
Factor \( \; y^2 x^4 \; \) out and add coefficients
\( \; 3 y^2 x^4 - 4 x^4 y^2 = y^2 x^4 ( 3 - 4) \; \)
\( \; = - y^2 x^4 \; \)

Add and Subtract Polynomials

You add polynomials by grouping like terms

Example 5

Add and simplify the two polynomials \( \; 3 x^2 + 2 x + 7 \; \) and \( \; 7 x^2 -9 x - 5 \; \)
Solution
Group like terms together
\( \; ( 3 x^2 - 2 x + 7 ) + ( 7 x^2 -9 x - 5 ) \; \)
\( \; = (3 x^2 + 7 x^2) +(- 2 x - 9 x) +(7 - 5) \; \)
and add them
\( \; = 10 x^2 - 11 x + 2 \; \)

Example 6

Add and simplify the two polynomials \( \; 3 x^2 + 2 x y + 7x - y^2 \; \) and \( \; -x^2 + 4 x y + 10x - 2 y^2 \; \)
Solution
Group like terms together
\( \; ( 3 x^2 + 2 x y + 7x - y^2 ) + ( -x^2 + 4 x y + 10x - 2 y^2 ) \; \)
\( \; = (3 x^2 - x^2) + (2 x y + 4 x y) + (7x + 10x) + (-y^2 -2y^2) \; \)
and add them
\( \; = 2 x^2 + 6 x y + 17x - 3y^2 \; \)

Example 7

Add , subtract and simplify:
\( \; ( 5 x^4 + 2 x^3 - 8 x^2 - 10 x + 2) + ( 7 x^3 - 9 x^2 - 5 x + 3 ) - ( - x^3 + 2 x^2 - 3 x + 7 ) \; \)
Solution
Remove parentheses and if a minus sign precedes it multiply all terms inside the parenthesis by \( - 1 \).
\( \; ( 5 x^4 + 2 x^3 - 8 x^2 - 10 x + 2) + ( 7 x^3 - 9 x^2 - 5 x + 3 ) - ( - x^3 + 2 x^2 - 3 x + 7 ) \\ = 5 x^4 + 2 x^3 - 8 x^2 - 10 x + 2 + 7 x^3 - 9 x^2 - 5 x + 3 + x^3 - 2 x^2 + 3 x - 7 \; \)
Group like terms
\( \; (5 x^4) + (2 x^3 + 7 x^3 + x^3 ) + (- 8 x^2 - 9 x^2 - 2 x^2 ) + ( - 10 x - 5 x + 3 x ) + ( 2 + 3 - 7 ) \\ = 5 x^4 + 10 x^3 - 19 x^2 -12 x - 2 \; \)

Questions

Expand (if necessary), ddd subtract and simplify the polynomials given below.
a) \( \; 2x - 3x + 3 y - y + 4 x - 5 y \; \) b) \( \; - 2(x - 3) - 4 (x + y + 2) - 5y \; \) c) \( \; 2x^2 - 3 x - 9 y^2 - y + 4 x - 5 y^2 - 5y \; \)
d) \( \; (-2x+4y-2) - 3(x- 6y-1) + 5 (x - y) \; \) e) \( \; (5x^2+4y-2) - 5(x - 6y^2 - 1) + 6 (2x - y) \; \)

Solutions

Expand (if necessary), ddd subtract and simplify the polynomials given below.
a) \( \; 3 x-3 y \; \) b) \( \; -6 x -9 y - 2\; \) c) \( \;2 x^2 -14 y^2 + x - 6 y \; \)
d) \( \; 17 y + 1 \; \) e) \( \; 5 x^2+30y^2 + 7 x - 2 y + 3 \; \)