# POLYNOMIAL EXPRESSIONS

Definition

 Terms with exactly the same unlike -denominators/cubic-equation-with-variable.html">variable parts (the same variables raised to the same powers ) are called like terms . Terms with Formulas /differential- equation - methods .html">different variable parts are called unlike terms.

Example 1
Simplify the polynomial ex pression 7x2 + 3xy − 4x2 + xy − y2 by combining like terms.

Solution

Example 2
Given x = -2 and y = 3, what is the value of xy 2 − 2xy .

Solution

Procedure

 Multiplying Polynomials Two polynomials are multiplied together by multiplying each term of the first polynomial by each term of the second polynomial and then summing the products .

Example 3
Find the product of (x2 − y)(2x + y2 − 3) .

Solution

Formulas

 Description Formula Difference of two squares Perfect Square Trinomials Sum of two cubes Difference of two cubes

Example 4
Factor 9x2 − 64y2.

Solution
Since the expression 9x2 − 64y2 is the difference of two squares we have
9x2 − 64y2
= (3x)2 − (8y)2
= (3x −8y)(3x + 8y)

Example 5
Factor 16m2 − 16mn2 + 4n4 .

Solution
16m2 − 16mn2 + 4n4
= 4[4m2 − 4mn2 + n4]
= 4[(2m)2 − 2 •(2m)(n2 ) + (n2 )2]
= 4(2m − n2 )2

§2-2 PROBLEM SET

Evaluate each polynomial for the given values.
1. 3x 2 + 2 where x = 1
2. − x2 where x = -2
3. 2x2 +3x − 3 where x = -2
4. 4t2 − 6t + 2 where t = 1/2
5. x2 y − 2x +1 where x = -1 and y = 3
6. 3ab − 2b2 + 2 where a = 2 and b = -2

Simplify by combining like terms.
7. (7x − 2) + (5x + 22)
8. (3y + 4) − (4y −12)
9. (3x2 − 2x + 7) + (x2 + 4x − 2)
10. (5x2 + 3x − 1) − (2x 2 − 4x − 2)
11. (2a2b + 5ab2 ) + (ab2 + 3ab − a2b)
12. (5b3 − ab + 4a2 ) − (a2 − b − ab)
13. (3x2 − 2x + 5) + (−3x2 + 2x + 1)
14. (xy2 + y − 2) − (x2 y + 2x + y)

Multiply and then simplify.
15. 4x2(2x − 9)
16. 4y2 (3y2 − 5y + 7)
17. 4a2 (5b2 − 3b + 4)
18. (2x + 3)(x − 1)
19. (x − 2)(x2 + 2x − 1)
20. 3(t − 6)(t 2 − t + 3)
21. (x − 5)(x + 1)2
22. (2r2 − r + 7)(r2 + 2r − 3)

Factor completely .
23. x2 − 9
24. 64r2 − 25s2
25. 4x2 +16
26. 98x3 − 50x
27. 9x2 − 60x +100
28. 25x2 + 40x +16
29. 27x4 + 72x3 + 48x2
30. 45a3 + 60a2 + 20a
31. 64x2 − 80xy + 25y2
32. 16a4 − 72a2b2 + 81b4
33. x2 + 4
34. x3 + 8
35. x3 − 27
36. 5x5 + 40x2
37. 4t4 − 32t
38. 3y3 + 27y
39. 64r3 + 27s3
40. a3 +8b6

§2-2 PROBLEM SOLUTIONS

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