5. Factor fully.
a)
2 2
3 2
x x x b)
2 5 4
3 2
x x x
c)
6 15 6 16 4 3 2
x x x x
d)
5 3 12 4
3 2
x x x
e)
6 46 15 3 2
x x x
f)
4 19 16 19 12 4 3 2
x x x x
g)
4 14 44 35 4 3 2
a) This expression cannot be factored further.
b) (2x + 5)(x + 4)
c) 2(x - 2)(x - 1)(x + 1)(x + 2)
d) (x + 2)(x - 3)(x + 4)
e) (2x + 3)(x - 5)(x + 2)
f) (x - 1)(x + 4)(x - 2)(x + 1)
g) 2(x + 1)(x + 2)(x - 7)
a) To factor fully: 2x^2 + 3x - 2x - 2
Step 1: Group the terms.
(2x^2 + 3x) - (2x + 2)
Step 2: Factor out common factors.
x(2x + 3) - 1(2x + 2)
Step 3: Factor out common binomial.
(x - 1)(2x + 2)
Therefore, the fully factored form is (x - 1)(2x + 2).
b) To factor fully: 2x^5 + 3x^2 + 4x^3 + 2x^2
Step 1: Group the terms.
(2x^5 + 4x^3) + (3x^2 + 2x^2)
Step 2: Factor out common factors.
2x^3(x^2 + 2) + x^2(3 + 2)
Step 3: Factor out common binomial.
2x^3(x^2 + 2) + x^2(5)
Therefore, the fully factored form is 2x^3(x^2 + 2) + 5x^2.
c) To factor fully: 6x^6 - 15x^4 - 6x^3 + 16x^2 + 4x + 3x^2 + 2
Since this expression involves multiple terms and there is no clear common factor, it cannot be factored further.
d) To factor fully: 5x^3 + 3x^2 - 12x^4 + 4x
Step 1: Group the terms.
(5x^3 - 12x^4) + (3x^2 + 4x)
Step 2: Factor out common factors.
x^3(5 - 12x) + x(3x + 4)
Therefore, the fully factored form is x^3(5 - 12x) + x(3x + 4).
e) To factor fully: 6x^4 + 46x^3 - 15x^2 + 3x^2
Step 1: Group the terms.
(6x^4 + 46x^3) - (15x^2 - 3x^2)
Step 2: Factor out common factors.
2x^3(3x + 23) - 3x^2(5)
Step 3: Factor out common binomial.
x^2(2x + 23)(3x - 5)
Therefore, the fully factored form is x^2(2x + 23)(3x - 5).
f) To factor fully: 4x^6 + 19x^4 - 16x^2 + 19x^2 + 12x + 4x^3 + 3x^2 + 2
Step 1: Group the terms.
(4x^6 + 19x^4 - 16x^2) + (19x^2 + 4x^3) + (12x + 2)
Step 2: Factor out common factors.
x^2(4x^4 + 19x^2 - 16) + x^2(4x + 19) + 2(6x + 1)
Step 3: Factor out common binomial.
x^2(4x^4 - 16 + 19x^2) + x^2(4x + 19) + 2(6x + 1)
Therefore, the fully factored form is x^2(4x^4 + 19x^2 - 16) + x^2(4x + 19) + 2(6x + 1).
g) To factor fully: 4x^6 + 14x^4 - 44x^2 + 35x^2 + 4x^3 + 3x^2 + 2
Step 1: Group the terms.
(4x^6 + 14x^4 - 44x^2) + (35x^2 + 4x^3) + 2
Step 2: Factor out common factors.
2x^2(2x^4 + 7x^2 - 22) + x^2(35 + 4x) + 2
Step 3: Factor out common binomial.
2x^2(2x^4 - 22 + 7x^2) + x^2(35 + 4x) + 2
Therefore, the fully factored form is 2x^2(2x^4 + 7x^2 - 22) + x^2(35 + 4x) + 2.
To factor fully expressions, we need to find the common factors between terms and group them together. Let's go through each expression and factor them one by one.
a) 2x^2 + 3x - 2x - 3
To factor this expression, we can group the terms:
(2x^2 + 3x) - (2x + 3)
Now, let's factor out the common terms from each group:
x(2x + 3) - 1(2x + 3)
Notice that we have a common factor of (2x + 3) in both groups. So, we can factor that out:
(2x + 3)(x - 1)
b) 2x^5 + 4x^3 + 3x^2
This expression does not have any common factors among the terms. So, we cannot factor it further.
c) 6x^15 - 6x^16 - 4x^3 + 6x^4 + 3x^2
Let's group the terms:
(6x^15 - 6x^16) - (4x^3 - 6x^4 - 3x^2)
Now, factor out the common terms from each group:
6x^15(1 - x) - 1x^2(4x^2 - 6x^3 - 3)
Again, we notice that both groups have a common factor of (1 - x). So, we factor that out:
(1 - x)(6x^15 - 1x^2(4x^2 - 6x^3 - 3))
d) 5x^3 + 12x^4 - 3x^2 + 4x^3
Group the terms:
(5x^3 + 12x^4) + (-3x^2 + 4x^3)
Factor out the common terms from each group:
x^3(5 + 12x) - x^2(3 - 4x)
There are no further common factors we can factor out.
e) 6x^46 - 15x^3 + 2x^2
No common factors among the terms, so we cannot factor it further.
f) 4x^19 - 16x^19 + 12x^4 + 19x^3 - 2x^2
Group the terms:
(4x^19 - 16x^19) + (12x^4 + 19x^3 - 2x^2)
Factor out common terms from each group:
(4 - 16x)(x^19) + (12x^4 + 19x^3 - 2x^2)
No further common factors to factor out.
g) 4x^14 + 44x^35 - 4x^3 + 2x^2
Group the terms:
(4x^14 + 44x^35) + (-4x^3 + 2x^2)
Factor out common terms from each group:
4x^14(1 + 11x^21) - 2x^2(2x - 1)
No additional common factors to factor out.
In conclusion, the fully factored expressions are:
a) (2x + 3)(x - 1)
c) (1 - x)(6x^15 - 1x^2(4x^2 - 6x^3 - 3))