- Polynomials
- General form of a polynomial in x:
-
anxn + an-1xn-1 +
an-2xn-2 + . . . +
a2x2 + a1x1 + a0, where
- the ai, i = 1, 2, . . ., n are real numbers
- n is a whole number
- Degree of a term is the sum of the exponents on the variables in the term.
- The term 4x3y5 has degree 8 since 3 + 5 = 8.
- Degree of a polynomial is the degree of the highest degree term.
- To write a polynomial in descending order for a certain variable means to write the
polynomial from the term with the highest exponent (in the certain variable) on the left
descending to the term with the lowest exponent (in the certain variable) on the right.
- Descending order in x: 3x2 - 2x + 1.
-
| Examples of Polynomials in x |
| Name |
Example |
Degree |
Note |
| Monomial |
3x2 |
2 |
One term (mono) |
| Binomial |
2x + 1 |
1 |
Two terms (bi) |
| Trinomial |
x3 + 2x2 + x |
3 |
Three terms (tri) |
| Polynomial |
6x4 + 5x3 + 4x2 + x + 7 |
4 |
Many terms (poly) |
- Polynomials can be in more than one variable....
-
| Examples of Polynomials in x and y |
| Name |
Example |
Degree |
Note |
| Monomial |
3x2y3 |
5 |
One term (mono) |
| Binomial |
2xy + y2 |
2 |
Two terms (bi) |
| Trinomial |
x3y4 + 2x3y + xy2 |
7 |
Three terms (tri) |
| Polynomial |
6x4y + 5x3y2 + 4x2y3 + xy4 + 7y5 |
5 |
Many terms (poly) |
- Adding Polynomials
- Combine like terms.
| Example: add 3x2 + 2x + 1 and 5x2 - 7x |
| (3x2 + 2x + 1) + (5x2 - 7x) = |
|
| = 3x2 + 2x + 1 + 5x2 - 7x |
remove parentheses |
| = 8x2 - 5x + 1 |
add like terms |
- Subtracting Polynomials
- 1) Remove parentheses (distribute "-" through).
- 2) Combine like terms.
| Example: subtract 3x2 + 2x + 1 from 5x2 - 7x |
| (5x2 - 7x) - (3x2 + 2x + 1) = |
|
| = 5x2 + 7x - 3x2 - 2x - 1 |
remove parentheses |
| = 2x2 + 5x - 1 |
add like terms |
- Multiplying Polynomials
- 1) Use distributive property to remove parentheses and multiply out.
- FOIL only works when multiplying binomials--the distributive property works when
multiplying any polynomials together.
- 2) Combine like terms.
| Example: (2x + 3)(4x + 5) |
| (2x + 3)(4x + 5) = |
|
| = 2x(4x + 5) + 3(4x + 5) |
distributive property |
| = (2x)(4x) + (2x)(5) + 3(4x) + 3(5) |
distributive property again |
| = 8x2 + 10x + 12x + 15 |
simplifying |
| = 8x2 + 22x + 15 |
combine like terms |
| Example: (2x - 3)(4x2 - 5x + 6) |
| (2x - 3)(4x2 - 5x + 6) = |
|
| = 2x(4x2 - 5x + 6) - 3(4x2 - 5x + 6) |
distributive property |
| = (2x)(4x2) - (2x)(5x) + (2x)(6) - 3(4x2) + 3(5x) - 3(6) |
distributive property again |
| = 8x3 - 10x2 + 12x - 12x2 + 15x - 18 |
simplifying |
| = 8x3 - 22x2 + 27x - 18 |
combine like terms |
- Special Forms
- (a + b)2 = a2 + 2ab + b2
- (a - b)2 = a2 - 2ab + b2
- (a + b)(a - b) = a2 - b2
- Note: the a and b may be any algebraic expression.
| Example |
Note |
| (x + y)2 = x2 + 2xy + y2 |
x is a
y is b |
(2x + 5)2 =
= (2x)2 + 2(2x)(5) + 52
= 4x2 + 20x + 25 |
2x is a
5 is b |
(2x - 5)2 =
= (2x)2 - 2(2x)(5) + 52
= 4x2 - 20x + 25 |
2x is a
5 is b |
(3x2 - 2y)2 =
= (3x2)2 - 2(3x2)(2y) + (2y)2
= 9x4 - 12x2y + 4y2 |
3x2 is a
2y is b |
(3x + 8)(3x - 8) =
= (3x)2 - 82 =
= 9x2 - 64 |
3x is a
8 is b |
(4x2 - 3y)(4x2 + 3y) =
= (4x2)2 - (3y)2
= 16x4 - 9y2 |
4x2 is a
3y is b |
25y2 - 81x4 =
= (5y)2 - (9x2)2
= (5y + 9x2)(5y - 9x2) |
5y is a
9x2 is b |
|
