ADDITION AND SUBTRACTION OF POLYNOMIALS
CHAPTER 4 SECTION 4
MTH 10905Algebra
Polynomial (“poly” means many)
Polynomial in x is an expression containing the sum of a finite number of terms of the form axn, any real number a and any whole number n
Polynomial are expressions not equations.
Expressions is a collection of numbers, letters, grouping symbols, and operations.
Equation shows that two expressions are equal.
Examples:
2x x2 – 2x + 1
43
1x
Polynomial
Answers should be in descending order (or descending powers) of the variable unless otherwise instructed.
2x4 + 4x2 - 6x + 3
Monomial is a Polynomial with one term.
Example:
8 because 8x0 4x because 4x1 -6x2
Polynomials
Binomial is a Polynomial with two terms.
Example:
x + 5 x2 – 6 4y2 – 5y
Trinomial is a Polynomial with three terms.
Example:
x2 – 2x +3 3z2 – 6z + 7
Degree of Term
Degree of Term of a polynomial in one variable is the exponent on the variable in that term
Example:4x2 Second2y5 Fifth
-5x First can be written -5x1
3 Zero can be written 3x0
Degree of Polynomial
Same as that of its highest-degree term
Example:
8x3 + 2x2 – 3x + 4 Third x2 - 4 Second6x - 5 First 4 Zero
x2y4 + 2X + 3 Sixth (sum of exponents)
More than 2 variables then add the exponent of highest degree
Add Polynomials
There are two ways to add polynomials: Horizontal expressions and Vertical (Column) form.
To add Polynomials combine the like terms.
Example:(7a2 + a – 6) + (10a2 – 3a + 9)
7a2 + a – 6 + 10a2 – 3a + 9 17a2 – 2a + 3
Add Polynomials
Example:(5x2 + 2x + y) + (x2 – 4x + 5)
5x2 + 2x + y + x2 – 4x + 56x2 – 2x + y + 5
Example:(5a2b + ab + 2b) + (7a2b – 3ab – b)
5a2b + ab + 2b + 7a2b – 3ab – b 12a2b – 2ab + b
Add Polynomials in Columns
Arrange the polynomial in descending order, one under the other with the like terms in the same column.
Add the terms in each column
Example:2
2
2
6 3 3
3 5
3 4 8
x x
x x
x x
Add Polynomials in Columns
Example: (4x3 + 3x – 4)+(4x2 – 5x – 7)
3
2
3 2
4 3 4
4 5 7
4 4 2 11
x x
x x
x x x
Subtract Polynomials
Use the distributive property to remove the parenthesis (this will change the signs in the second polynomial)
Combine like terms
Subtract Polynomials
Example:(4x2 – 3x + 6) - (x2 – 7x + 8) 4x2 – 3x + 6 - x2 + 7x – 8 4x2 – x2 – 3x + 7x + 6 – 8 3x2 + 4x – 2
Subtract Polynomials
Do we represent “subtract a from b” as a – b or b – a?
b - a
Example:Subtract (-x2 – 4x + 2) from (x3 + 3x + 9)
(x3 + 3x + 9) - (-x2 – 4x + 2) x3 + 3x + 9 + x2 + 4x – 2 x3 + x2 + 3x + 4x + 9 – 2 x3 + x2 + 7x + 7
Subtract Polynomials in Columns
Write the polynomial being subtracted below the polynomial from which it is being subtracted. List the like terms in the same column.
Change the sign of each term in the polynomial being subtracted.
Add the terms in each column.
Subtract Polynomials in Columns
2123
45
674
2
2
2
xx
xx
xx
Example:
Subtract (x2 – 5x + 4) from (4x2 + 7x + 6)
Subtract Polynomials in Columns
3734
8 3
5 7 4
23
2
3
xxx
x
xx
Example:
Subtract (3x2 – 8) from (-4x3 + 7x – 5)
Remember
When adding drop the parentheses and combine the like terms.
When subtracting use the distributive property to change the signs in the second polynomial.
You can only evaluate and simplify a polynomial because they are expressions. You can NOT solve a polynomial because it is not an equation.
HOMEWORK 4.4
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