1. Algebraic Expressions
A variable
A monomial is an expression of the form axk, where a is a real number and k is a nonnegative integer.
A binomial

2. Algebraic Expressions
Note that the degree of a polynomial is the highest power of the variable that appears in the polynomial.

3. Example 1– Adding and Subtracting Polynomials
(a) Find the sum (x3 – 6x2 + 2x + 4) + (x3 + 5x2 – 7x).
(b) Find the difference (x3 – 6x2 + 2x + 4) – (x3 + 5x2 – 7x).
Solution: (a) (x3 – 6x2 + 2x + 4) + (x3 + 5x2 – 7x)
= (x3 + x3) + (–6x2 + 5x2) + (2x – 7x) + 4
= 2x3 – x2 – 5x + 4
Group like terms
Combine like terms

4. Example 1 – Solution (b) (x3 – 6x2 + 2x + 4) – (x3 + 5x2 – 7x)
cont’d
(b) (x3 – 6x2 + 2x + 4) – (x3 + 5x2 – 7x)
= x3 – 6x2 + 2x + 4 – x3 – 5x2 + 7x
= (x3 – x3) + (– 6x2 – 5x2) + (2x + 7x) + 4
= –11x2 + 9x + 4
Distributive Property
Group like terms
Combine like terms

5. Example 3 – Multiplying Polynomials
Find the product: (2x + 3) (x2 – 5x + 4)
Solution 1: Using the Distributive Property
(2x + 3)(x2 – 5x + 4)
= 2x(x2 – 5x + 4) + 3(x2 – 5x + 4)
= (2x  x2 – 2x  5x + 2x  4) + (3  x2 – 3  5x + 3  4)
= (2x3 – 10x2 + 8x) + (3x2 – 15x + 12)
Distributive Property
Distributive Property
Laws of Exponents

6. Special Product Formulas

7. Example 4 – Using the Special Product Formulas
Use the Special Product Formulas to find each product.
(a) (x2 – 2)3
Solution: (a) Substituting A = x2 and B = 2 in Product Formula 5, we get (x2 – 2)3 = (x2)3 – 3(x2)2(2) + 3(x2)(2)2 – 23
= x6 – 6x4 + 12x2 – 8

8. Example 9 – Recognizing the Form of an Expression
Factor each expression.
(a) x2 – 2x – 3 (b) (5a + 1)2 – 2(5a + 1) – 3
Solution:
(a) x2 – 2x – 3 = (x – 3)(x + 1)
(b) This expression is of the form
where represents 5a + 1. This is the same form as the expression in part (a), so it will factor as
Trial and error

9. Example 9 – Solution
cont’d
= (5a – 2)(5a + 2)

10. Special Factoring Formulas
Some special algebraic expressions can be factored using the following formulas.
The first three are simply Special Product Formulas written backward.

11. Example 11 – Factoring Differences and Sums of Cubes
Factor each polynomial.
(a) 27x3 – (b) x6 + 8
Solution: (a) Using the Difference of Cubes Formula with A = 3x and B = 1, we get
27x3 – 1 = (3x)3 – 13 = (3x – 1)[(3x)2 + (3x)(1) + 12]
= (3x – 1) (9x2 + 3x + 1)

12. Example 11 – Solution
cont’d
(b) Using the Sum of Cubes Formula with A = x2 and B = 2, we have
x6 + 8 = (x2)3 + 23
= (x2 + 2)(x4 – 2x2 + 4)

13. Example 12 – Recognizing Perfect Squares
Factor each trinomial.
(a) x2 + 6x (b) 4x2 – 4xy + y2
Solution:
(a) Here A = x and B = 3, so 2AB = 2  x  3 = 6x. Since the middle term is 6x, the trinomial is a perfect square.
By the Perfect Square Formula we have
x2 + 6x + 9 = (x + 3)2

14. Example 12 – Solution
cont’d
(b) Here A = 2x and B = y, so 2AB = 2  2x  y = 4xy. Since the middle term is –4xy, the trinomial is a perfect square.
By the Perfect Square Formula we have
4x2 – 4xy + y2 = (2x – y)2

15. Special Factoring Formulas
When we factor an expression, the result can sometimes be factored further. In general, we first factor out common factors, then inspect the result to see whether it can be factored by any of the other methods of this section. We repeat this process until we have factored the expression completely.

16. Example 13 – Factoring an Expression Completely
Factor each expression completely.
(a) 2x4 – 8x2 (b) x5y2 – xy6
Solution: (a) We first factor out the power of x with the smallest exponent.
2x4 – 8x2 = 2x2(x2 – 4)
= 2x2(x – 2)(x + 2)
Common factor is 2x2
Factor x2 – 4 as a difference of squares

17. Example 13 – Solution
cont’d
(b) We first factor out the powers of x and y with the smallest exponents.
x5y2 – xy6 = xy2(x4 – y4)
= xy2(x2 + y2)(x2 – y2 )
= xy2(x2 + y2)(x + y)(x – y)
Common factor is xy2
Factor x4 – y4 as a difference of squares
Factor x2 – y2 as a difference of squares

18. Example 15 – Factoring by Grouping
Factor each polynomial.
(a) x3 + x2 + 4x (b) x3 – 2x2 – 3x + 6
Solution:
(a) x3 + x2 + 4x + 4 = (x3 + x2) + (4x + 4)
= x2(x + 1) + 4(x + 1)
= (x2 + 4)(x + 1)
Group terms
Factor out common factors
Factor out x + 1 from each term

19. Example 15 – Solution (b) x3 – 2x2 – 3x + 6 = (x3 – 2x2) – (3x – 6)
cont’d
(b) x3 – 2x2 – 3x + 6 = (x3 – 2x2) – (3x – 6)
= x2(x – 2) – 3(x –2)
= (x2 – 3)(x – 2)
Group terms
Factor out common factors
Factor out x – 2 from each term