2nd Year Algebra Reviewer 1a7256

Factoring Polynomials To factor a polynomial means to write the polynomial as a product of other polynomials. A. Greatest Common Factor Examples: 1. 4x3y2 + 12 x3y + 20 xy2 2. -12x4 + 16x3 + 8x2 3. x2n + xn+1 + xn , n >0 B. Factor by Grouping The Distributive Property is used to factor a common binomial factor from an expression. Examples: 4. 4a(2b + 3) - 5(2b + 3) 5. 6r(r - s) - 7(s - r) Some polynomials can be factored by grouping so that a common binomial factor is found. 6. 8y2 + 4y – 6ay – 3a 7. 3x(y - 4) - 2(4 - y) 8. xy – 4x – 2y + 8 C. Factor trinomials of the form x2 + bx + c 9. x2 - 5x - 24 10. x2 + 8x + 12 11. x2 + 5x - 84 12. x2 + 7xy + 12y2 Not all trinomials can be factored using only integers. Consider x2 - 3x – 6. This trinomial is not factorable. Why? The polynomial x2 - 3x – 6 is a prime polynomial, and it is nonfactorable over the integers.

D. Factor trinomials of the form ax2 + bx + c 13. 2x2 - 21x + 10 14. 3x2 + 11x + 8 15. 4z2 – 17z – 21 16. 3x2 - 11x + 4 17. 3y2 - y + 8 18. 10 - 17x - 6x2

A polynomial is factored completely when it is written as a product of factors that are nonfactorable over the integers. 19. 4x3 + 12x2 – 160x

20. 12x3y + 14x2y – 6xy

21. 30y + 2xy – 4x2y

22. x 2 a  6  14 x a 3 y a  2  72 y 2 a  4 4c 4  15c 2  9

23.

y 2 a  30 y a  64

24. x 2 a  6  14 x a 3 y a  2  72 y 2 a  4

SPECIAL FACTORING A.

Difference of Two Squares.

1. 4x2 – 9 2. 25r2 – 36t6 3. c2 – 9d4 4. 100y8 - 49 B.

Sum and Difference of Two Cubes

a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2) 1. x3 – 8 2. 27x3 + 1 3. 125y6 – 64 4. 27x12 – 216 C.

Perfect-Square Trinomial

a2 + 2ab + b2 = (a + b) 2 a2 – 2ab + b2 = (a – b) 2 1. 4x2 - 12x + 9 2. x2 + 12x + 36 3. 4t2 – 12t + 9 4. 25y2 + 70y + 49

SOLVING QUADRATIC EQUATIONS Standard Form of the Quadratic Equation ax 2  bx  c  0 Principle of Zero Product – If the product of 2 factors equals 0 then at least one of the factors equals 0. Double Root – If a quadratic equation has 2 solutions that are the same then the solution is called a double root. Solve by factoring:

1. x2  8x  15  0

3. 4x2 – 9x = 0

2. x 2  10 x  25  0

4. 2x2 - 21x = - 10

Solve using the square root method:

1.

x 2  64

3.

( x + 2 )2 = 17

2.

6x2 = 25

4.

( 2x – 3 )2 + 4 = 0

Solve using the completing the square method:

1.

x2 + 8x + 12 = 0

3.

3y2 - 4y + 8 = 0

2.

2x2 - 6x + 10 = 0

4.

2y2 + 15y – 35 = 27 + 3y

Solve using the quadratic formula: 1. x 2  10x  24  0

2. 3x 2  5x  2  0

PROBLEM SOLVING. 1. One number is 5 greater than another number. If their product is 234, find the two numbers.

2. The length of a rectangular shaped lawn is 3 times as long as its width. If its area is 432 m2. Find the length and width of the lawn.

3. The width of a rectangle is 15cm less than its length. The area of the rectangle is 286cm2. Find the dimensions of the rectangle.