(1) Find the Gcf of the Expressions 12X3 and 18X4

Exercises:

(1) Find the gcf of the expressions 12x3 and 18x4.

Exercises:

(2) Factor out the gcf:

(a)

(b)

(3) Factor by grouping:

(a) rx + 2rs + 2x + 4s.

(b) 3xy + 6y -2x - 4.

(4) Factor:

(a) x2 - 9,

(b) 7x2 - 700,

(5) Factor the polynomial:

(a)

(b) .

(c)

(d)

(e) .

(6) Factor the polynomial:

(a) 6x2 – x – 15

(b) 5x2 – 7x – 6

(7) Factor the polynomial:

(a)

(b)

(c)

(8) Factor the polynomial:

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(9) Solve the polynomial equation:

(a)

(b)

(c)

Solutions:

(1) the gcf of the coefficients 12 and 13 is 6. The lowest power of x is x3. So the

GCF = 6x3.

(2)

(a) = (the gcf = 5x).

5x(x2 + 2x - 3)

5x3 10x2 -15x

5x2 5x2 5x2

(b) (gcf = 4xy)

4xy(xy - 2y + 3).

4x2y2 -8xy2 12xy

4xy 4xy 4xy

(3) (a) rx + 2rs + 2x + 4s. (pair the factors and find the gcf)

gcf = r gcf = 2 (factor out the gcf from the pairs)

(combine like terms on top of (x + 2s) by

adding the coefficients r and 2)

(b) 3xy + 6y -2x - 4 (pair off the terms and find the gcf:)

gcf = 3y gcf = -2 (Note that I needed to make the signs match by using

gcf = -2 in the second pair. Now pull out the gcf's:)

(Now combine like terms on (x + 2) )

(4) (a) x2 - 9, (DOS )

(x)2 - 32 (split it up with the formula)

= (x + 3)(x - 3).

(b) 7x2 - 700 = (gcf = 7, factor it out:)

7(x2 - 100) = (DOS in pharentheses).

(x)2 - 102

7(x - 10)(x + 10).

2(x4 - 1) = (rewrite x4 - DOS in pharentheses:)

(x2)2 - 12, (split it up using formula:)

= (x2 - 1 is a DOS - split it up again)

(the complete factorization).

(5) (need two numbers that multiply to 7 add to 8 - there ain't

many choices since 7 is prime - only 1 and 7 are factors)

1 7 (1 + 7 = 8, so:)

= (x + 1)(x + 7).

(b) (need two numbers which * to -15, add to -2

list the factors of -15. The larger ones will have to be

1 -15 negative.)

3 -5 (3 + -5 = -2, so: )

= (x + 3)(x - 5).

has to be positive this time, )

-1 20

-2 10

-4 5 (-4 + 5 = 1)

= (x - 4)(x + 5).

(d) (need two numbers which * to 12, add to -8 - both factors will have

to be negative to add to a negative, * to a positive)

-1 -12

-2 -6 (-2 + -6 = -8)

-3 -4

= (x - 2)(x - 6).

(e) = (first, factor out the gcf = 7)

7(x2 + 2xy - 3y) = (find the two numbers which * to -3, + to 2)

-1 3 (add a y to the -1 and 3:)

7(x - y)(x + 3y).

(6) (a) 6x2 – x – 15 (multiply the lead and constant terms)

(6)(-15) = -90

1 -90

2 -45

3 -30

5 -18

6 -15

9 -10 (9 + -10 = -1 - use these to split up -x:)

= 6x2 + 9x - 10x - 15 (pair the terms and get the gcf's)

gcf = 3x gcf = -5 (picked - 5 to make signs match)

= 3x(2x + 3) - 5(2x + 3) (factor out gcf's, combine terms: )

= (3x - 5)(2x + 3). (voila. finito. )

(b) 5x2 – 7x – 6 (Multiply the lead and constant terms:)

5(-6) = -30,

1 -30

2 -15

3 -10 (3 + -10 = -7 - use them to split up -7x)

5x2 - 10x + 3x - 6 (pair off, gcf's:)

gcf = 5x gcf = 3 (factor gcf's, combine terms: )

5x(x - 2) + 3(x - 2) =

(5x + 3)(x - 2).

(8)(-21) = -168

-1 168

-2 84

-3 56

-4 42

-6 28 (-6 + 28 = 22 : use them to break up 22x)

8x2 + 28x - 6x - 21 = (pair off, find gcf's, make signs match)

gcf = 4x gcf = -3 (factor out gcf's)

4x(2x + 7) - 3(2x + 7) = (combine like terms:)

(4x - 3)(2x + 7). (rock-action)

(7)

(a) (This is a DOC)

(x)3 - 103 (use formula with a = x, b = 10).

(x - 10)(x2 + 10x + 100).

a - b a2 ab b2

(b) (This is a SOC)

(3x)3 + (4)3 = (use formula, with a = 3x, b = 4).

(3x + 4)(9x2 - 12x + 16) (answer)

(3x)2 (3x)(4) (4)2

5(x3 - 1) (DOC inside pharentheses, w/ a = x, b = 1)

(yeah)

(8)(a) (no gcf, trinomial, lead coeff = 1)

-1 -25 (need two numbers which * to 25, + to -10:

-5 -5 -5 + -5 = -10 )

= (x - 5)(x - 5).

(b) (gcf = 4x, factor it out: .)

= no more factoring can be done)

= (DOS in pharentheses (x)2 - 32)

(complete factorization)

(d) (no gcf, trinomial, lead coeff = 3, grouping)

3*4 = 12

1 -12 (use 1 and -12 to break up 11x)

= (pair off, gcf's)

gcf = x, gcf = -4 (signs have to match)

= (combine terms)

(e) x3 + 216 = (no gcf, two terms, 6 is the cube of 216, so SOC)

(x)3 + (6)3 ( x plays the role of a, 6 plays the role of b)

a + b a2 -ab + b2

(f) (no gcf, four terms - grouping, pair off, gcf's)

gcf = 4x gcf = 7 (factor out gcf's)

(combine terms.)

(g) (gcf = 6, factor it out:)

(two terms, DOS in pharentheses (x)2 - 22 )

(split it up - complete factorization)

(h) (no gcf, two terms, note the cube x3, and 1000 = 103, 27 = 33)

(10x)3 - 33 (DOC, w/ 10x playing a, 3 playing b)

(10x)2 3*10x 32

(i) (no gcf, trinomial, lead coefficient = 1)

1 30

2 15

3 10

5 6 ( 5 + 6 =11)

= (x + 5)(x + 6).

(j) (no gcf, trinomial, lead coefficient ¹1, grouping:)

(2)(-15)=-30

-1 30

-2 15

-3 10 (-3 + 10 = 7 - use them to break up 7x)

= (pair off, gcf's)

gcf = x gcf = 5 (pull out gcf's)

= (combine terms:)

(9) (a) (LHS = 0, so, factor LHS as a DOS)

(x + 5)(x - 5) = 0, (set factors = 0,)

x + 5 = 0, x - 5 = 0,

-5 -5 +5 +5

x = 5, x = -5. (give answer in solution set: )

x = {±5}

(b) (get LHS = 0)

-6 -6

(factor LHS)

(set factors = 0)

+3 +3 -2 -2

x = 3, x = -2,

x = {3, -2}

-28x -28x

(gcf = 2x)

(inside pharentheses - trinomial, lead coefficient = 1)

-1 14

-2 7 (-2 + 7 = 5)

(factors = 0)

x= 0 -7 -7 +2 +2,

x = 0, x = -7, x = 7.

x = {0,±7}