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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