Chapter 5

Factori ng

5.1 FACTORING

The factors of a given algebraic expression consist of two or more algebraic expressions which when multiplied together produce the given expression.

EXAMPLES 5.1. Factor each algebraic expression.

(a )

(6) (c) (d)

X* - 7~ + 6 = (X - l)(x - 6) x2 + Bu = X(X + 8) 6x2 - 7~ - 5 = ( 3 ~ - 5 ) ( b + 1) x2 + 2xy - 8y2 = (X + 4y)(x - 2y)

The factorization process is generally restricted to finding factors of polynomials with integer coefficients in each of its terms. In such cases it is required that the factors also be polynomials with integer coefficients. Unless otherwise stated we shall adhere to this limitation.

Thus we shall not consider (x - 1) as being factorable into (fi + l)(fi - 1) because these factors are not polynomials. Similarly, we shall not consider (x2 - 3y2) as being factorable into (x - f i y ) (x + f i y ) because these factors are not polynomials with integer coefficients. Also, even though 3x + 2y could be written 3(x + 5) we shall not consider this to be a factored form because x + ;y is not a polynomial with integer coefficients.

A given polynomial with integer coefficients is said to be prime if it cannot itself be factored in accordance with the above restrictions. Thus x2 - 7x + 6 = (x - l)(x - 6) has been expressed as a product of the prime factors x - 1 and x - 6.

A polynomial is said to be factored completely when it is expressed as a product of prime factors.

Note 1 . In factoring we shall allow trivial changes in sign. Thus x2 - 7x + 6 can be factored either as (x - l)(x - 6) or (1 - x)(6 - x). It can be shown that factorization into prime factors, apart from the trivial changes in sign and arrangement of factors, is possible in one and only one way. This is often referred to as the Unique Factorization Theorem.

Note 2. Sometimes the following definition of prime is used. A polynomial is said to be prime if it has no factors other than plus or minus itself and +1. This is in analogy with the definition of a prime number or integer such as 2,3,5,7,11, . . . and may be seen to be equivalent to the previous definition.

Note 3. Occasionally we may factor polynomials with rational coefficients, e.g., x2 - 9/4 = (x + 3/2)(x - 3/2). In such cases the factors should be polynomials with rational coefficients.

Note 4. There are times when we want to factor an expression over a specific set of numbers, e.g., x2 - 2 = (x + t/z)(x - t/z) over the set of real numbers, but it is prime over the set of rational numbers. Unless the set of numbers to use for the coefficients of the factors is specified it is assumed to be the set of integers.

5.2 FACTORIZATION PROCEDURES

In factoring, formulas I-VIII of Chapter 4 are very useful. Just as when read from left to right they helped to obtain products, so when read from right to left they help to find factors.

The following procedures in factoring are very useful. 31

32 FACTORING [CHAP. 5

A. Common monomial factor. Type: ac + ad = a(c + d ) EXAMPLES 5.2. (a ) &u2y - 2x3 = P ( 3 y - x )

(b) 2r3y - xy2 + 3x2y = xy(2x2 - y + 3 ~ )

B. Difference of two squares. Type: a2 - b2 = (a + b)(a - b )

EXAMPLES 5.3. (a ) x2 - 25 = x2 - 52 = ( x + 5) (x - 5 ) where a = x , 6 = 5 (b) 4x2 - 9y2 = (2x)2 - ( 3 ~ ) ~ = (2r + 3y)(2r - 3y) where a = 2x, b = 3y

C. Perfect square trinomials. Types: a2 + 2ab + b2 = (a + b)2 a2 - 2ab +- b2 = (a - b)2

It follows that a trinomial is a perfect square if two terms are perfect squares and the third term is numerically twice the product of the square roots of the other two terms.

EXAMPLES 5.4. (a ) x2 + 6x + 9 = ( x + 3)2 (b) 9x2 - l h y + 4y2 = ( 3 ~ - 2 ~ ) ~

D. Other trinomials. Types: 2 + (a + b)x + ab = ( x + a)(x + b ) acx2 + (ad + bc)x + bd = (ax + b)(cx + d)

EXAMPLES 5.5. (a) x2 - 5x + 4 = (x - 4)(x - 1) where a = -4, b = -1 so that their sum (a + b) = -5 and their product ab = 4.

(6 ) 2 + xy - 12y2 = (x - 3y)(x + 4y) where a = -3y, 6 = 4y (c ) 3x2 - 5x - 2 = (x - 2)(3x + 1). Here ac = 3, bd = -2 , ad + bc = -5 ; and we

find by trial that a = 1, c = 3, b = -2, d = 1 satisfies ad + bc = -5 .

(e) 8 - 14x + 5x2 = (4 - 5x)(2 - x ) (d) 62 + x - 12 = (3x - 4)(2r + 3) '

E. Sum, difference of two cubes. Types: a3 + b3 = (a + b)(a2 - ab + b2) a3 - b3 = (a - b)(a2 + ab + b2)

EXAMPLES 5.6. (a) @ + 27y3 = ( 2 ~ ) ~ + ( 3 ~ ) ~ = (2 + 3Y)"2d2 - (W(3Y) + (3Y)21 = (2r + 3y)(4x2 - 6xy + 9y2)

(b) 8x3y3 - 1 = - l3 = ( b y - 1)(4x2y2 + b y + 1)

F. Grouping of terms. Type: ac + bc + ad + bd = c(a + b) + d(a + b) = ( a + b)(c + d)

EXAMPLE 5.7. 2ax - 4bx + ay - 2by = 2x(a - 2b) + y(a - 26) = (a - 2b)(2x + y)

G. Factors of a" k 6". Here we use formulas VII and VIII of Chapter 4.

EXAMPLES 5.8. (a) 32x5 + 1 = (2)s + 15 = (2r + 1)[(2)4 - (2r)3 + (2)~ - 2x + 11 = (2X + 1)(1k4 - 8x3 + 4x2 - 2X + 1)

(b) x 7 - l = ( x - l ) ( x 6 + . 9 + x 4 + x 3 + x 2 + x + l )

CHAP. 51 FACTORING 33

H. Addition and subtraction of suitable terms.

EXAMPLE 5.9. Factor x4 + 4. Adding and subtracting 4 2 (twice the product of the square roots of x4 and 4), we have

x4 + 4 = (x4 + 4 2 + 4) - 4x2 = (x2 + 2)2 - (&)2 = (2 + 2 + &)(x2 + 2 - 2u) = (x2 + 2 x + 2)(x2 - 2 J + 2 )

I . Miscellaneous combinations of previous methods.

EXAMPLES 5.10. (a) x4 - xy3 - W3y + y4 = (x4 - xy3) - (x3y - y4) = 4 x 3 - y3) - y(x3 - y3) = (x3 - y3)(x - y ) = (x - y ) ( 2 + xy + y2)(x - y ) = ( x - y)2 (2 + xy + y2)

(6) x2y - 3x2 - y + 3 = (x2y - 3x2) + ( - y + 3 ) = 20, - 3) - (y - 3 ) = 0, - 3 ) ( 2 - 1) = (y - 3)(x + l)(x - 1)

(c ) x 2 + 6 x + 9 - y 2 = ( x 2 + & y + 9 ) - y 2 = ( x + 3 ) 2 - y 2

= [ (x + 3 ) + Y 1 [ (x + 3 ) - Y 1 = ( x + y + 3 ) ( x - y + 3 )

5.3 GREATEST COMMON FACTOR

The greatest common factor (GCF) of two or more given polynomials is the polynomial of highest degree and largest numerical coefficients (apart from trivial changes in sign) which is a factor of all the given polynomials.

The following method is suggested for finding the GCF of several polynomials. (a) Write each polynomial as a product of prime factors. (6) The GCF is the product obtained by taking each factor to the lowest power to which it occurs in any of the polynomials.

EXAMPLE 5.1 1. The GCF of 23 32(x - (x + 2 ~ ) ~ . 22 33(x - y) (x + 2 ~ ) ~ , 32(x - y ) 2 ( x + 2 y ) is 32(x - Y ) ~ (x + 2Y).

Two or more polynomials are relatively prime if their GCF is 1.

5.4 LEAST COMMON MULTIPLE

The least common multiple (LCM) of two or more given polynomials is the polynomial of lowest degree and smallest numerical coefficients (apart from trivial changes in sign) for which each of the given polynomials will be a factor.

The following procedure is suggested for determining the LCM of several polynomials. (a ) Write each polynomial as a product of prime factors. (b) The LCM is the product obtained by taking each factor to the highest power to which it occurs.

EXAMPLE 5.12. The LCM of 2332(x - Y ) ~ ( x + 2 ~ ) ~ . 2233(x - y)* (x + Z Y ) ~ , 32(x - Y ) ~ ( x + 2y) is 2333(x - (x + 2y)3.

34 FACTORING

Solved Problems

[CHAP. 5

Common Monomial Factor

Type: ac + ad = a(c + d ) 5.1 (U) 2r2 - 3xy = x(2r - 3y)

(6) 4x + 8y + 122 = 4(x + 2y + 32) (cl 3 2 + &r3 + 1 2 r 4 = ~ ( i + 2r + 4 2 ) ( d )

( e )

9s3 t + 15s2P - 3s2tZ = 3s2t(3s + 58 - I ) 10a2 b3 c4 - 15a3 b2 c4 + 30a4 b3c2 = 5a2 62 c?(26c2 - 3 a 2 + 6a2 6)

(f) 4~"" - 8a2" = 40"+'(l- 2fl-I)

Difference of Two Squares

Type: a2 - b2 = (a + b)(a - 6 ) 5.2 (U)

(6) (c) 9x2y2- 16u2 = (3xy)2 - (&)2 = (3xy + 4a)(3xy - 4a) (d) 1 - m2n4 = l2 - (mn2)2 = (1 + mn2)(1 - mn2) (e) 3x2 - 12 = 3(x2 - 4) = 3(x + 2)(x - 2)

(g ) x4 - y4 = (x2)2 - (y2)2 = (2 + y2)(x2 - y2) = (2 + y2)(x + y)(x-y) (h) 1 - x8 = (1 + x4)(1 - x4) = (1 + x4)(1 + x2)(i - x 2 ) = (1 + x4)(1 + x2)(i + x)(i - x)

0') x3y - y3x = xy(x2 - y2) = xy(x + y)(x - y)

( f ) (5x + 2y)2 - (3x - 7y)2 = [(5x + 2y) + (3x - 7y)][(5x + 2y) - (3x - 7y)l = (ax - 5y) (h + 9y)

x2 - 9 = x2 - 32 = (X + 3)(x - 3) 25x2 - 4y2 = ( 5 ~ ) ~ - ( 2 ~ ) ~ = ( 5 ~ + 2y)(Sx - 2y)

(f) x2y2 - 36y4 = y2[x2 - ( 6 ~ ) ~ ] = y2(x + 6y)(x - 6y)

(i) 3 2 ~ ~ 6 - 162P = 26(16a4 - 81b4) = 2b(4a2 + 962)(4~2 - 9b2) = 26(4a2 + 9b2)(2a + 36)(2~ - 36)

(k) ( X + 1)2 - 36y2 = [ (x + 1) + (6y)] [ ( x + 1) - (6y)] = ( X + 6y + l)(x - 6y + 1)

Perfect Square Trinomials

Types: a2 -k 2ab + b2 = (a + bl2 a2 - 2ab + b2 = (a - b)'

5.3 (a) x2 + 8x + 16 = x2 + 2(x)(4) + 42 = (x + 4)2 (6) 1 + 4y + 4y2 = (1 + 2 ~ ) ~ (c ) t2 - 4t + 4 = ? - 2(t)(2) + 22 = (t - 2)2 ( d ) x2 - 1 6 ~ ~ + 64y2 = (X - 8 ~ ) ~ (e) 25x2 + 6Oxy + 36y2 = (5x + 6 ~ ) ~ (f) 16m2 - 40mn + 25n2 = (4m - 5n)2 (g) 9x4 - 2 4 ~ ~ ~ + 16y2 = (3x2 - 4 ~ ) ~ ( h ) 2 ~ 3 ~ 3 + 1 6 ~ 2 ~ 4 + 32ry5 = 2ry3(2 + ky + 16~2) = 2ry3(x + 4y)2 (i) 0')

(f)

16u4 - 72a2b2 + 81b4 = (4a2 - 962)2 = [(2a + 36)(2a - 36)12 = (2a + (x + 2 ~ ) ~ + 1O(x + 2y) + 25 = (x + 2y + 5)2

4m6n6 + 32m4n4 + 64m2n2 = 4m2n2(m4n4 + 8m2n2 + 16) = 4m2n2(m2n2 + 4)2

- 36)2

(k) a2x2 - 2abxy + 62y2 = (U -

CHAP. 51 FACTORING 35

Other Trinomials

Types: x2 + (a + b)x + ab = ( x + a)(x + b ) acx2 +- (ad + bc)x + bd = (ax -+ b)(cx + d)

5.4 (a) x2 + 6x + 8 = (x + 4)(x + 2) ( 6 )

(c) ( d )

x2 - 6x + 8 = ( X - 4)(x - 2) x2 + 2x - 8 = ( X + 4 ) ( ~ - 2) x2 - 2x - 8 = ( X - 4)(x + 2)

(e) x2 - 7xy + 12y2 = (x - 3y)(x - 4y) (f) x2 + xy - 12y2 = (x + 4y)(x - 3y) (g)

(h )

( i )

16 - 1 0 ~ + x2 = (8 - x)(2 - X ) 20 - x - x2 = (5 + x)(4 - X ) 3x3 - 3 2 - 18x = 34x2 - x - 6) = 3x(x - 3)(x + 2)

0') y4 + 7y2 + 12 = (y2 + 4)(y2 + 3) ( k ) m4 + m2 - 2 = (m2 + 2)(m2 - 1) = (m2 + 2)(m + l ) (m - 1) ( I ) (x + 1)2 + 3(x + 1) + 2 = [(x + 1) + 2 ] [ ( x + 1) + 11 = (x + 3)(x + 2) (m) s2? - 2FP - 63t4 = ?(s2 - 2Ft - 6312) = ?(s - 9t)(s + 7t) (n) t4 - l o t2 + 9 = (t2 - i ) (z2 - 9) = (t + i ) ( t - i ) ( t + 3)(t - 3) (0) 2x6y - 6x4y3 - &2y5 = 2X2y(x4 - 32y2 - 4y4)

(p) x2 - b y +y2 + 1O(x - y ) + 9 = (x -y)2 + 1O(x - y ) + 9

(4) 4x'y'O - 40x5y7 + 84x2y4 = 4X2y4(x6y6 - 10x3y3 + 21) = 4x2y4(x3y3 - 7)(x3y3 - 3) ( r ) x ~ - Y - ~ O = ( X ~ - ~ ) ( P + ~ )

(s) xm+2n + 7 P + " + 10xm = x m ( P + 7x" + 10) = xrn(xn + 2)(x" + 5 )

= 2X2y(x2 + y2)(x2 - 4y2) = 2X2y(x2 + y2)(x + 2y)(x - 2y)

= [ ( x - y ) + l ] [ ( x - y ) + 9 ] = ( x - y + l ) ( x - y + 9 )

2) ( t ) a2b-1) - 5 d - 1 + 6 = (ay-1 - 3)(aY-' -

3x2 + 1ox + 3 = (3x + l ) ( x + 3) 2x2 - 7x + 3 = (2x - l ) (x - 3) 2y2 - y - 6 = (2y + 3 ) b - 2) 10s2 + 11s - 6 = (5s - 2 ) ( 2 + 3) 6 x 2 - xy - 12y2 = (3x + 4y)(2K - 3y) 10 - x - 3x2 = (5 - 3 ~ ) ( 2 + X ) 4t4 - 9 t 2 + 2 = (t2 - 2)(4z2 - 1) = (z2 - 2 ) ( 2 ~ + 1 ) ( 2 ~ - 1) 16x3y + 28r2y2 - 3 0 ~ ~ ~ = b y ( @ + 1 4 ~ ~ - 1 5 ~ ' ) = h y ( 4 ~ - 3 y ) ( h + 5y) 12(x + Y ) ~ + 8(x + y ) - 15 = [6(x + y ) - 5 ] [ 2 ( ~ + y ) + 31 = (6x + 6y - 5 ) ( h + 2y + 3) 6b2"+l + 56"+' - 66 = b(6b2" + 56" - 6) = b(26" + 3)(36" - 2) l8X4P+'" - 66x2P+my2 - 24xmy4 = 6.P(3x4P - 11x*y2 - 4y4) = 6P(3x2P + y2)(x2P - 4y2)

= 6.P(3x2P + y2)(.r" + 2y)(xP - 2y) 64xI2y3 - 68x8y7 + 4x4y' = 4x4y3( 16X8 - 17x4y4 + y8) = 4x4y3( 1&r4 - y4)(x4 - y4)

= 4x4y3(42 + y2)(4x2 - y2) (2 + y2)(x2 - y2) = 4x4y3(42 + y2)(2X + y)(2x-y)(x2 + y2)(x + y)(x - y )

Sum of Difference of Two Cubes

Types: a3 + b3 = (a + b)(a2 - ab + b2) a3 - b3 = (a - b)(a2 + ab + b2)

36 FACTORING [CHAP. 5

5.6 (a) x3 + 8 = 2 + Z3 = (x + 2)(x2 - 2x + 22) = ( x + 2)(x2 - 2x + 4) ( 6 ) a3 - 27 = a3 - 33 = (U - 3)(a2 + 3~ + 32) = (U - 3)(a2 + 3~ + 9) (c) u6 + b6 = ( u ~ ) ~ + (b2)3 = (0' + b2)[(a2)2 - a2b2 + (b2)2]

= (a2 + b2)(a4 - a2b2 + b4) ( d ) a6 - b6 = (a3 + b3)(a3 - b3) = (U + b)(a2 - ab + b2)(a - b)(a2 + ab + b2) (e) u9 + b9 = ( u ~ ) ~ + (b3)3 = (a3 + b3)[(a3)2 - a3b3 + (b3)2]

(f) a12 + 612 = (a4)3 + (b4)3 = (a4 + b4)(a8 - a464 + b8) ( g ) 64x3 + 1 2 5 ~ ~ = ( 4 ~ ) ~ + ( 5 ~ ) ~ = ( 4 ~ + 5 ~ ) [ ( 4 ~ ) ~ - ( 4 ~ ) ( 5 y ) + ( S Y ) ~ ]

= (U + b)(a2 - ab + b2)(a6 - a3b3 + b6)

= ( 4 ~ + Sy)( 16K2 - 2 0 ~ ~ + 2 5 ~ ~ ) ( h ) ( x + y ) 3 - z3 = ( X + y - z) [ (x + y ) 2 + ( x + y ) z + 22]

(i) (x - 2)3 + sY3 = (x - 213 + (2y)3 = (x - 2 + 2y)[(x - 2)2 - (x - 2)(2y) + (2y )2~ = (x - 2 + 2y)(x2 - 4x + 4 - 2xy + 4y + 4y2)

= ( x + y - z)(x2+ 2xy +y2+xz + y z + 22)

0') x6 - 7x3 - 8 = (x3 - 8)(2 + 1 ) = (x3 - ~ 3 ) ( ~ 3 + 1) = (x - 2 ) ( 2 + 2x +

= 2 y ( x + 2y)@ - b y + 4y2)(x - 2y)(x2 + 2xy + 4y2)

+ i)(2 - + 1) (k) x8y - ~ 2 ~ 7 = x2y(x6 - ~ ~ 6 ) = x2y(x3 + ~ ~ 3 ) ( ~ 3 - ~ ~ 3 ) = xzy[x3 + (2y )3~ - ~ ~ 1 3 1

(0 54x6y2 - 3&r3y2 - 16y2 = 2y2(27x6 - 19x3 - 8) = 2y2(27x3 + 8)(x3 - 1 ) = 2 ~ ~ [ ( 3 ~ ) ~ + 2 3 ] ( ~ 3 - 1 ) = 2y2(3x + 2)(9x2 - &r + 4 ) ( ~ - 1)(x2 + x + 1 )

Grouping of Terms

Type: ac + bc +ad +- bd = c(a + b) + d(a +- b) = (a +- b)(c+ d ) 5.7 ( U )

( b )

(c)

bX - ab + x2 - M = b(x - a) + X ( X - a) = ( X - a)( b + X ) = ( X - a)(x + b) ~ U X - UY - 3 6 ~ + by = ~ ( 3 x - y ) - b(3x - y ) = ( ~ x - Y ) ( u - 6 ) 6x2 - 4~ - 9 b ~ + 6ab = b ( 3 x - 2 ~ ) - 3 b ( 3 ~ - 2 ~ ) = ( 3 ~ - 2 ~ ) ( 2 x - 36)

(d) a x + a y + x + y = a ( x + y ) + ( x + y ) = ( x + y ) ( u + 1 ) (e) x2 - 4y2 + x + 2y = (x + 2y)(x - 2y) + (x + 2y) = (x + 2y)(x - 2y + 1 ) (f) 2 + x2y + xy2 + y3 = x2(x + y ) + y2(x + y ) = ( x + y ) ( 2 + y2) ( g ) X' + 27x4 - x3 - 27 = x4(x3 + 27) - (x3 + 27) = (x3 + 27)(x4 - 1)

= (x3 + 33)(x2 + 1)(2 - 1 ) = (x + 3 ) ( 2 - 3x + 9 ) ( 2 + l ) ( X + l ) ( x - 1 )

= ( x - 1 ) ( 2 + x + 1)(y + 2)@2 - 2y + 4) (h) x3y3 - y3 + &r3 - 8 = y3(x3 - 1 ) + 8(x3 - 1) = (x3 - l)(y3 + 8)

) (i) a6 + b6 - a2b4 - a4b2 = u6 - a2b4 + b6 - a4b2 = a2(a4 - b4) - b2(a4 - b4 = (a4 - b4)(a2 - b2) = (a2 + b2)(a2 - b2)(a + b)(a - b) = (a2 + b2)(n + 6)(a - 6)(a + b)(a - b) = (a2 + b2)(a + b)2(a - b)2

(j) a3 + 3a2 - 5ab + 262 - b3 = (d - b3) + (3a2 - 5ab + 2b2) = (U - b)(a2 + ab + b2) + (U - b ) ( 3 ~ - 26) = (U - b)(a2 + ab + b2 + 3~ - 26)

Factors of a" k 6"

5.8 an + 6" has a + b as a factor if and only if n is a positive odd integer. Then a" + b" = (a +- b)(a"-' - a"-2b + an-3b2 - . . . - abn-2 + bn-1).

(a) a3 + 63 = (a + b)(a2 - a6 + b2) (6) 64 + y3 = 43 + y3 = (4 + ~ ) ( 4 ~ - 4y + y2) = (4 + y)( 16 - 4y + y2)

CHAP. 51 FACTORING 37

(c) x3 + 8y6 = x3 + ( 2 ~ ~ ) ~ = (x + 2y2)[x2 - 4 2 3 ) + ( 2 ~ ' ) ~ ] = (x + 2y2)(x2 - b y 2 + 4y4)

(d) U' + 6' = (U + 6)(a4 - a3 b + a2 b2 - ab3 + b4) (e ) 1 + x5y5 = i s + (xy)' = (1 + x y ) ( l - xy + x2y2 - 2 y 3 + x4y4) (f) z5 + 32 = z5 + 25 = ( z + 21(~4 - 2z3 + 22z2 - 23z + 24)

(g)

= (z + 2)(z4 - 2z3 + 4z2 - 82 + 16) + x l o = ( 2 ) s + ( 2 1 5 = (a2 + X2)[(a2)4 - (u2)3x2 + (u2)2(X2)2 - (a2)(X2)3 + (x2)41

= (a2 + X*)(U8 - a6x2 + u4x4 - u2x6 + 2) (h) U7 + U7 = (U + .)(U6 - u5u + U 4 3 - U 3 3 + u2u4 - uv5 + u6) (i) x9 + 1 = ( x ~ ) ~ + I3 = (x3 + 1)(x6 - x3 + 1) = (x + l)(x2 - x + 1)(x6 - x3 + 1)

5.9 - b" has a - b as a factor if n is any positive integer. Then an - bn = (a - b)(an-' + an-2b + an-3 b + 8 . + abn-2 + bn-').

If n is an even positive integer, an - bn also has a + b as factor. ( U )

(6) (c)

a2 - 62 = (U - b)(a + 6) a3 - b3 = (U - 6)(a2 + a6 + b2) 27x3 - y3 = ( 3 ~ ) ~ - y3 = ( 3 ~ - Y ) [ ( ~ x ) ~ + (3x)y + y2] = ( 3 ~ - y)(9x2 + 3xy + y2)

(e ) us - 32 = us - 2' = (a - 2)(a4 + a3.2 + a2.22 + a.23 + 24)

(f) y7 - 27 = (y - z)(y6 + y5z + y4z2 + y3z3 + y2z4 + yz5 + 26) (g) x6 - a6 = (x3 + a3)(x3 - a3) = (x + n ) ( 2 - ax + a2)(x - a ) ( 2 + ax + 2) (h) U* - U* = (U4 + u4)(u4 - u4) = (U4 + u4)(u2 + u2)(u2 - 2)

(i) x9 - 1 = (x313 - 1 = (x3 - 0') x l o - y l o = (x5 + y5)(x5 - y5)

(d) ~ - x ~ = ( ~ - x ) ( ~ ~ + ~ x + x ~ ) = ( ~ - x ) ( ~ + x + x ~ )

= (a - 2)(a4 + 2u3 + 4a2 + 8a + 16)

= (U4 + u4)(u2 + u2)(u + u)(u - U)

= (x + y)(x4 - x3y + x2y2 - xy3 + y4)(x - y)(x4 + x3y + x2y2 + xy3 + y4)

+ x3 + 1) = (x - 1)(x2 + x + 1)(x6 + x3 + 1)

Addition and Subtraction of Suitable Terms

5.10 (a) a4 + a2b2 + 64 (adding and subtracting a2b2) = (a4 + 2 ~ ~ 6 ~ + 64) - a2b2 = (a2 + b2)2 - ( ~ 6 ) ~

= (a2 + b2 + &)(a2 + b2 - ab) (6) 36x4 + 15x2 + 4 (adding and subtracting 9x2)

= ( 3 6 ~ ~ + 24x2 + 4) - 9x2 = ( 6 ~ ~ + 2)2 - ( 3 ~ ) ~ = [(h2 + 2) + 3x1 [(6x2 + 2) - 3x1 = (sX2 + 3~ + 2)(6x2 - 3x + 2)

(c) 64x4 + y4 (adding and subtracting 16x23) = ( 6 4 ~ ~ + 1&r2y2 + y4) - 1&W2y2 = (8~' + y2)2 - ( ~ x Y ) ~

= (8x2 + y2 + 4xy)(&2 + y2 - 4xy) (d) u8 - 14u4 + 25 (adding and subtracting 4u4)

= - 1oU4 + 25) - 4u4 = - - (2u2)2 = (u4 - 5 + 2u2)(u4 - 5 - zU2) = (.4 + 2u2 - ~ ) ( ~ 4 - 2 2 - 5 )

Miscellaneous Problems

5.11 ( U ) x2 - 4z2 + 9y2 - 6xy = (x2 - 6xy + 9y2) - 4z2 = (x - 3y)2 - (2z)2 = (x - 3y + 2z)(x - 3y - 22)

38 FACTORING [CHAP. 5

(b) 16a2 + lObc - 253 - b2 = 16a2 - (b2 - lObc + 2 5 ~ ~ ) = ( 4 ~ ) ~ - (b - 5 ~ ) ~ = ( 4 ~ + b - 5 ~ ) ( 4 ~ - b + 5 ~ )

(c) x 2 + 7x +y2- 7y - b y - 8 = (2 - 2xy +y2) + 7(x -y) - 8 = ( X - Y ) ~ + 7 ( ~ - y ) - 8 = ( X - y + 8 ) ( ~ - y - 1)

(d) a2 - 8ab - 2uc + 16b2 + 8bc - 1 5 9 = (a2 - 8ab + 16b2) - (2ac - 86c) - 1 5 3

( e ) m4 - n4 + m3 - mn3 - n3 + m3n = (m4 - mn3) + (m3n - n4) + (m3 - n3) = m(m3 - n3) + n(m3 - n3) + (m3 - n3) = (m3 - n3)(m + n + 1) = (m - n)(m2 + mn + n2)(m + n + 1)

= (U - 4b)2 - ~ C ( U - 46) - 15c2 = (U - 46 - %)(U - 46 + 3 ~ )

Greatest Common Factor and Least Common Multiple

5.12 (a ) 9x4y2 = 3 ~ ~ 2 , 12x3~3 = ~ - 3 ~ 3 ~ 3 GCF = 3x3y2, LCM = 22 - 3 2 ~ 4 y 3 = 3h4y3

(b) 48?t4 = 24.3?t4, 549t6 = 2 ~ 3 ~ ? P , 60r49 = 22-3.5r4$ GCF=2-3?$=6?$, LCM=24-33-5r4t6=2160r4t6 6x - 6y = 2 . 3 ( ~ - U), GCF = 2 ( ~ - y ) , y4 - 16 = (r2 + 4)0, + 2 ) b - 2), GCF = y - 2,

(c) 4x2 - 4y2 = 2 2 ( ~ 2 - y2) = 2 2 ( ~ + Y)(X - y) LCM = 22 * 3(x + Y)(X - y)

( d ) y2 - 4 = (r + 2)0, - 2), y2 - 3y + 2 = 0, - 1)(r - 2) LCM = b2 + 4 ) b + 2)(r - 2)(y - 1)

(e) 3.52(x + 3 ~ ) ~ ( 2 x - y ) ~ , 23-32.5(x+3y)3(2x-y)2, 22.3-5(x+ 3 ~ ) ~ ( 2 x - y ) ~ GCF = 3.5(x+ 3y)2(2x -y)2, LCM = 2 3 . 3 2 q x + 3y)4(2x-y)5

Supplementary Problems

Factor each expression.

& 5.13 (a) 3x2y4 + 6x3y3 (b) 12.~~9 - &t4 + 4s41 (c) 2X2yz - 4xyz2 + axy2z3

(e) 1 -a4 (f) 6 4 x - x 3

(d) 4y2- 100

(g) 8r4- 128

5.14 (a ) m4-4m2-21

(b) a4 - 20a2 + 64 (c) 4s4t - 4s3? - 24s2?

x ~ + ~ + 5X"'+4 - 50x4 (d)

(h) 18r3y-8ry3 (0) 3a4 + 6a2b2 + 3b4 (i) 0') 4(x + 3y)2 - 9(2x - y)2 (4) x2 + 7x + 12 ( k ) x 2 + 4 x + 4 ( r ) y2-4y - 5

(m) x2y2 - 8xy + 16

( 2 ~ + y)2 - (3y - z12 (p) (m2 - n2)2 + 8(m2 - n2) + 16

(0 4- 12y+9y2 (s) x2 - &JJ + 15y2 ( t ) 2z3 + 1oz2 - 282

(n) 4x3y + 1b2y2 + 9xy3 (U) 15+2x-x2 ( e ) 2x2+3x+ 1 (i) 36z6 - 13z4 + z2 (f) 3y2 - l ly + 6 (g) 5m3 - 3m2 - 2m ( k ) 4x2n+2 - 4xn+2 - 3x2

0') 12(x - y)2 + 7(x - y) - 12

(h ) &r2 + 5xy - 6y2

& 5.15 (a) y3+27 (d) 8z4-27z7 (g) Y 6 + 1 (b) x 3 - 1 ( e ) k 4 y - 64xy4 ( h ) (x - 2)3 + 0, + I ) ~ (c) x3y3+8 (f) m9-n9 (i) 8x6 + 7x3 - 1

a 5 . 1 6 (a) xy+3y-2x-6 (c) a2 + bx - ux - b ( d ) x3 - xy2 - x2y + y3

(e) z7 - 2z6 + z4 - 2t3 (f) m3 - mn2 + m2n - n3 + m2 - n2 (6) 2pr - ps + 6qr - 3qs

CHAP. 51 FACTORING 39

5.17 (a ) z5+ 1 ( b ) x5+32y5 (c) 32-U' (d) mlo- 1 ( e ) 1 -

5.18 (a) z4+ 64 (4 m2 - 4p2 + 4mn + 4n2 (f) 9x2 - x2y2 + 4y2 + 1 2 ~ y (b) 4x4 + 3x2y2 + y4 (e) 6ub + 4 - a2 - 9b2 (c) x 8 - 12x4 + 16

( g ) x2 + y2 - 422 + 2xy + 3xz + 3yz

& 5.19 Find the GCF and LCM of each group of polynomials. (a) 16y2z4, 24y3z2

(b) 9?s2?, 12?s4?, 21?s2

(c) (d) 6y3 + 1 2 ~ ~ 2 , 6y2 - 24z2, 4y2 - ~ Y Z - 24z2

x2 - 3xy + 2y2, 4x2 - 16Xy + 16y2

(e ) x 5 - x , x 5 - x 2 , x 5 - x 3

ANSWERS TO SUPPLEMENTARY PROBLEMS

(m2 - 7)(rn2 + 3) (4 (2 + l ) ( x + 1) (i) ~ ~ ( 2 2 + 1 ) ( 2 ~ - 1 ) ( 3 ~ + 1)(32 - 1 ) (a + 2)(a - 2)(a + 4)(a - 4) 4s2t(s - 3t)(s + 29 ( g ) m(5m + 2)(m - 1 ) ( k ) x2(h" + l ) ( z m - 3) x4(P - 5 ) ( P + 10)

0, + 3)b2 - 3y + 9) (x - 1)(x2 + x + 1) (xy + 2)(x2y2 - 2xy + 4)

&y(x - 2y)(x2 + 2ry + 4y2)

(2r - S M + 3 d

(Z + i)(z4 - z3 + - + 1)

(f) (3Y - 2 ) b - 3)

(h) (2x + 3Y)(3X - 2Y)

(f) (m - n)(m2 + mn + n2)(rn6 + m3n3 + n6) (g) (Y2 + MY4 - Y 2 + 1) (h) (x + y - l)(x2--xy + y 2 - 5x + 4y + 7 )

0') (4x - 4y - 3)(3x - 3y + 4)

z4(2 - 3 ~ ) ( 4 + 62 + 9z2) (i) (h - 1)(4x2 + 2x + l)(x + 1)(x2 - x + 1)

(x + 3)(y - 2) (c ) (ax + b)(x - 1 ) ( e ) z3(z - 2)(2 + l)(z2 - z + 1 ) (4 (x - Y I 2 ( X + Y ) (f) (m + n)(m - n)(m + n + 1 )

(x + 2y)(x4 - 2x3y + 4x2y2 - &y3 + 16y4) (2 - U)( 16 + 8u + 4u2 + 2u3 + u4) (m + I ) (m4- m3 + m2 - m + I)(m - l ) (m4 + m3 + m2 + m + 1 ) (1 - ~ ) ( i + z + z2 + z3 + r4 + z5 + z6) (z2 + 42 + 8)(z2 - 42 + 8) (e ) (2 + a - 3b)(2 - a + 36) (2x2 + XY + y2)(2r2 - XY + y2) (f) ( 3 ~ + XY + 2 y ) ( 3 ~ - XY + 2y) (x4 + 2x2 - 4)(x4 - 2x2 - 4) (g) (x +Y + 4 4 ( x +Y - 4

(d) (m + 2n + 2p)(m + 2n - 2p)

40 FACTORING

5.19 (a) GCF = 23y2z2 = 8y2z2, LCM = 24-3y3z4 = 48y3z4 (b) GCF = 3?s2, LCM = 252rSs4rS (c) ( d ) (e)

GCF = x - 2y, GCF = 20, + 22), GCF = x(x - l) ,

LCM = 4 ( ~ - Y)(X - 2 ~ ) ~ LCM = 1 2 ~ ~ 0 , + 22)b - 22)0, - 32) LCM = X ~ ( X + I ) (x - l)(x2 + l ) ( 2 + x + 1)

[CHAP. 5

80a8c7fa_Cover0e4e30c9_3192a14c35_326a22f795_3337efafeb_347c7638b1_3566424221_36c0a5b3a4_373fada8cc_38b8c036fe_395f5eb3d9_40