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Exercises:
(1) Find the gcf of the expressions 12x3 and 18x4.
Exercises:
(2) Factor out the gcf:
(a) 5 x3  10 x 2  15 x
(b) 4 x 2 y 2  8xy 2  12 xy
(3) Factor by grouping:
(a) rx + 2rs + 2x + 4s.
(b) 3xy + 6y -2x - 4.
(4) Factor:
(a) x2 - 9,
(b) 7x2 - 700,
(c) 2x4 - 2.
(5) Factor the polynomial:
(a) x 2  8 x  7.
(b) x 2  2 x  15 .
(c) x 2  x  20.
(d) x 2  8 x  12.
(e) 7 x 2  14 xy  21y .
(6) Factor the polynomial:
(a) 6x2 – x – 15
(b) 5x2 – 7x – 6
(c) 8x2 + 22x – 21
(7) Factor the polynomial:
(a) x3  1000
(b) 27 x3  64
(c) 5 x 3  5
(8) Factor the polynomial:
(a) x 2  10 x  25.
(b) 4 x 2  36 x.
(c) 4 x3  36 x.
(d) 3 x 2  11x  4.
(e) x3  216.
(f) 4 x 2  8 x  7 x  14.
(g) 6 x 2  24
(h) 1000 x3  27
(i) x 2  11x  30.
(j) 2 x 2  7 x  15.
(9) Solve the polynomial equation:
(a) x 2  25  0.
(b) x 2  x  6.
(c) 2 x3  10 x 2  28 x.
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) 5 x3  10 x 2  15 x =
5x(x2 + 2x - 3)
5x3 10x2
5x2 5x2
(the gcf = 5x).
-15x
5x2
(b) 4 x 2 y 2  8xy 2  12 xy
4xy(xy - 2y + 3).
(gcf = 4xy)
4x2y2 -8xy2 12xy
4xy 4xy 4xy
(3) (a) rx + 2rs + 2x + 4s.
gcf = r
gcf = 2
r ( x  2s)  2( x  2s)
(r  2)( x  2 s )
(b) 3xy + 6y -2x - 4
gcf = 3y gcf = -2
3 y ( x  2)  2( x  2)
(3 y  2)( x  2) .
(4) (a) x2 - 9,
(x)2 - 32
(pair the factors and find the gcf)
(factor out the gcf from the pairs)
(combine like terms on top of (x + 2s) by
adding the coefficients r and 2)
(pair off the terms and find the gcf:)
(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) )
(DOS )
(split it up with the formula)
= (x + 3)(x - 3).
(b) 7x2 - 700 =
7(x2 - 100) =
(x)2 - 102
7(x - 10)(x + 10).
(gcf = 7, factor it out:)
(DOS in pharentheses).
(c) 2x4 - 2 =
2(x4 - 1) =
(x2)2 - 12,
2( x 2  1)( x 2  1) =
2( x  1)( x  1)( x 2  1)
(gcf = 2, factor it out:)
(rewrite x4 - DOS in pharentheses:)
(split it up using formula:)
(x2 - 1 is a DOS - split it up again)
(the complete factorization).
(5) x 2  8 x  7.
1 7
= (x + 1)(x + 7).
(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 = 8, so:)
(b) x 2  2 x  15 (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).
(c) x 2  x  20. (need two numbers which * to -20, add to +1 - the larger factor
has to be positive this time, )
-1 20
-2 10
-4 5 (-4 + 5 = 1)
= (x - 4)(x + 5).
(d) x 2  8 x  12. (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) 7 x 2  14 xy  21y =
7(x2 + 2xy - 3y) =
-1 3
7(x - y)(x + 3y).
(first, factor out the gcf = 7)
(find the two numbers which * to -3, + to 2)
(add a y to the -1 and 3:)
(6) (a) 6x2 – x – 15
(6)(-15) = -90
(multiply the lead and constant terms)
1 -90
2 -45
3 -30
5 -18
6 -15
9 -10
= 6x2 + 9x - 10x - 15
(9 + -10 = -1 - use these to split up -x:)
(pair the terms and get the gcf's)
gcf = 3x gcf = -5
= 3x(2x + 3) - 5(2x + 3)
= (3x - 5)(2x + 3).
(picked - 5 to make signs match)
(factor out gcf's, combine terms: )
(voila. finito. )
(b) 5x2 – 7x – 6
5(-6) = -30,
(Multiply the lead and constant terms:)
1 -30
2 -15
3 -10
2
5x - 10x + 3x - 6
(3 + -10 = -7 - use them to split up -7x)
(pair off, gcf's:)
gcf = 5x gcf = 3
5x(x - 2) + 3(x - 2) =
(5x + 3)(x - 2).
(factor gcf's, combine terms: )
(c) 8x2 + 22x – 21
(8)(-21) = -168
(multiply constant and lead term:)
-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
4x(2x + 7) - 3(2x + 7) =
(4x - 3)(2x + 7).
(factor out gcf's)
(combine like terms:)
(rock-action)
(7)
(a) x3  1000
(This is a DOC)
3
3
(x) - 10
(use formula with a = x, b = 10).
(x - 10)(x2 + 10x + 100).
a - b a2
ab
b2
(b) 27 x3  64
(This is a SOC)
(3x)3 + (4)3 =
(use formula, with a = 3x, b = 4).
2
(3x + 4)(9x - 12x + 16)
(answer)
(3x)2 (3x)(4) (4)2
(c) 5 x 3  5
5(x3 - 1)
5( x  1)( x2  x  1).
(8)(a) x 2  10 x  25.
-1 -25
-5 -5
= (x - 5)(x - 5).
(b) 4 x  36 x.
2
= 4 x( x  9)
(c) 4 x3  36 x. =
4 x( x 2  9) =
4 x( x  3)( x  3).
(gcf = 5, take it out: )
(DOC inside pharentheses, w/ a = x, b = 1)
(yeah)
(no gcf, trinomial, lead coeff = 1)
(need two numbers which * to 25, + to -10:
-5 + -5 = -10 )
4 x2
36 x
(gcf = 4x, factor it out:
 x,
 9 .)
4x
4x
no more factoring can be done)
(gcf = 4x, factor it out:)
(DOS in pharentheses (x)2 - 32)
(complete factorization)
(d) 3 x 2  11x  4.
3*4 = 12
(no gcf, trinomial, lead coeff = 3, grouping)
1 -12
= 3x  x  12 x  4
(use 1 and -12 to break up 11x)
(pair off, gcf's)
2
gcf = x, gcf = -4
(signs have to match)
= x(3x  1)  4(3 x  1) (combine terms)
= ( x  4)(3 x  1).
(e) x3 + 216 =
(x)3 + (6)3
( x  3)( x 2  3x  9).
a + b a2 -ab + b2
(no gcf, two terms, 6 is the cube of 216, so SOC)
( x plays the role of a, 6 plays the role of b)
(f) 4 x 2  8 x  7 x  14.
(no gcf, four terms - grouping, pair off, gcf's)
gcf = 4x gcf = 7
4 x( x  2)  7( x  2) 
(4 x  7)( x  2).
(factor out gcf's)
(combine terms.)
(g) 6 x 2  24
(gcf = 6, factor it out:)
6( x 2  4) 
6( x  2)( x  2).
(two terms, DOS in pharentheses (x)2 - 22 )
(split it up - complete factorization)
(h) 1000 x3  27
(no gcf, two terms, note the cube x3, and 1000 = 103, 27 = 33)
3
3
(10x) - 3
(DOC, w/ 10x playing a, 3 playing b)
2
(10 x  3)(100 x  30 x  9) .
(10x)2 3*10x 32
(i) x 2  11x  30.
(no gcf, trinomial, lead coefficient = 1)
1 30
2 15
3 10
5
6
= (x + 5)(x + 6).
(j) 2 x 2  7 x  15.
(2)(-15)=-30
( 5 + 6 =11)
(no gcf, trinomial, lead coefficient 1, grouping:)
-1 30
-2 15
-3 10
(-3 + 10 = 7 - use them to break up 7x)
2
= 2 x  3x  10 x  15  (pair off, gcf's)
gcf = x
gcf = 5
(pull out gcf's)
= x(2 x  3)  5(2 x  3)  (combine terms:)
( x  5)(2 x  3).
(9) (a) x 2  25  0.
(x + 5)(x - 5) = 0,
x + 5 = 0, x - 5 = 0,
-5 -5 +5 +5
x = 5, x = -5.
x = {5}
(b) x 2  x  6.
-6 -6
2
x  x  6  0,
( x  3)( x  2)  0,
x  3  0, x  2  0.
+3 +3 -2 -2
x = 3, x = -2,
x = {3, -2}
(c) 2 x3  10 x 2  28 x.
(LHS = 0, so, factor LHS as a DOS)
(set factors = 0,)
(give answer in solution set: )
(get LHS = 0)
(factor LHS)
(set factors = 0)
(get LHS = 0)
-28x -28x
2 x  10 x  28 x  0
2 x( x 2  5x  14)  0,
3
2
(gcf = 2x)
(inside pharentheses - trinomial, lead coefficient = 1)
-1 14
-2 7
(-2 + 7 = 5)
2 x( x  7)( x  2)  0,
(factors = 0)
2 x  0, x  7  0, x  2  0,
x= 0 -7 -7 +2 +2,
x = 0, x = -7, x = 7.
x = {0,7}