Question

factor completely
1.) $3v^3 - 2v^2 + 21v - 14$
a) $(v^2 - 7)(v^2 + 2)$
b) $(3v + 7)(v^2 + 2)$
c) $(v^2 + 7)(3v - 2)$
d) $(v^2 - 7)(3v - 7)$
2.) $18b^3 + 3b^2 - 48b - 8$
a) $(3b^2 - 8)(6b + 1)$
b) $(3b^2 + 8)(3b^2 - 1)$
c) $(3b^2 + 8)(6b - 1)$
d) $2(3b^2 + 8)(3b + 4)$
3.) $8x^4 + 8x^3 + 40x^2 + 40x$
a) $8x(x^2 - 5)(x + 1)$
b) $(x + 5)(x^2 + 1)$
c) $8x(x^2 + 5)(x + 1)$
d) $8x(x^2 + 5)(x - 1)$
4.) $4a^3 + 256$
a) $4(a + 4)(a^2 + 4a + 16)$
b) $4(a + 4)^3$
c) $4(a + 4)(a^2 - 4a + 16)$
d) $4(a - 4)(a^2 + 4a + 16)$
5.) $64 + 125x^3$
a) $(4 - 5x)(16 + 20x + 25x^2)$
b) $(4 + 5x)(4 - 5x)^2$
c) $(4 + 5x)(16 - 20x + 25x^2)$
d) $(4 + 5x)^3$
6.) $192x^3 + 3y^3$
a) $3(4x - y)(16x^2 - 4xy + y^2)$
b) $3(4x + y)(16x^2 - 4xy + y^2)$
c) $3(4x - y)(16x^2 + 4xy + y^2)$
d) $3(4x + y)^3$
7.) $16a^3 - 250b^3$
a) $2(2a - 5b)(4a^2 + 10ab + 25b^2)$
b) $2(2a - 5b)^3$
c) $2(2a - 5b)(2a + 5b)$
d) $2(2a + 5b)(4a^2 - 10ab + 25b^2)$
8.) $375x^3 - 81$
a) $3(5x - 3)^3$
b) $3(5x - 3)(5x + 3)$
c) $3(5x + 3)(25x^2 - 15x + 9)$
d) $3(5x - 3)(25x^2 + 15x + 9)$
9.) $32 - 108x^3$
a) $4(2 + 3x)(4 - 6x + 9x^2)$
b) $4(2 - 3x)(2 + 3x)$
c) $4(2 - 3x)^3$
d) $4(2 - 3x)(4 + 6x + 9x^2)$

Explanation:

Problem 1: \( 3v^3 - 2v^2 + 21v - 14 \)

Step 1: Group Terms

Group the first two and last two terms: \( (3v^3 - 2v^2) + (21v - 14) \)

Step 2: Factor Each Group

Factor \( v^2 \) from the first group: \( v^2(3v - 2) \)
Factor \( 7 \) from the second group: \( 7(3v - 2) \)

Step 3: Factor Out Common Binomial

Now we have \( v^2(3v - 2) + 7(3v - 2) \), factor out \( (3v - 2) \): \( (v^2 + 7)(3v - 2) \)

Step 1: Group Terms

Group as \( (18b^3 + 3b^2) + (-48b - 8) \)

Step 2: Factor Each Group

Factor \( 3b^2 \) from first group: \( 3b^2(6b + 1) \)
Factor \( -8 \) from second group: \( -8(6b + 1) \)

Step 3: Factor Out Common Binomial

Now we have \( 3b^2(6b + 1) - 8(6b + 1) \), factor out \( (6b + 1) \): \( (3b^2 - 8)(6b + 1) \)

Step 1: Factor Out GCF

First, factor out \( 8x \): \( 8x(x^3 + x^2 + 5x + 5) \)

Step 2: Group and Factor

Group the polynomial inside the parentheses: \( (x^3 + x^2) + (5x + 5) \)
Factor \( x^2 \) from first group: \( x^2(x + 1) \)
Factor \( 5 \) from second group: \( 5(x + 1) \)
Now we have \( x^2(x + 1) + 5(x + 1) \), factor out \( (x + 1) \): \( (x^2 + 5)(x + 1) \)

Step 3: Combine with GCF

Putting it all together: \( 8x(x^2 + 5)(x + 1) \)

Answer:

C) \( (v^2 + 7)(3v - 2) \)

Problem 2: \( 18b^3 + 3b^2 - 48b - 8 \)