Question

week 1: factoring polynomials
factor each polynomial completely.

1. $x^2 + x - 2$
2. $x^2 - 5x - 6$
3. $x^2 - 5x + 6$

1. $x^2 - xy - 6y^2$
2. $24 + 5x - x^2$
3. $x^2 - 13x + 42$

1. $3x^2 - 5x + 2$
2. $2x^2 - x - 1$
3. $5x^2 + 26x + 5$

1. $12x^2 + 7x + 1$
2. $-9x^2 + 3x + 2$
3. $9x^2 - 4y^2$

1. $49 - 9x^2$
2. $x^2 + 49$
3. $x^3 - 64$
4. $5x^3 + 40$

Explanation:

Step1: Factor $x^2+x-2$

Find two numbers: $2$ and $-1$ (sum $1$, product $-2$).
$x^2+x-2=(x+2)(x-1)$

Step2: Factor $x^2-5x-6$

Find two numbers: $-6$ and $1$ (sum $-5$, product $-6$).
$x^2-5x-6=(x-6)(x+1)$

Step3: Factor $x^2-5x+6$

Find two numbers: $-2$ and $-3$ (sum $-5$, product $6$).
$x^2-5x+6=(x-2)(x-3)$

Step4: Factor $x^2-xy-6y^2$

Find two terms: $-3y$ and $2y$ (sum $-y$, product $-6y^2$).
$x^2-xy-6y^2=(x-3y)(x+2y)$

Step5: Factor $24+5x-x^2$

Rearrange and factor: $-(x^2-5x-24)$. Find two numbers: $-8$ and $3$ (sum $-5$, product $-24$).
$24+5x-x^2=-(x-8)(x+3)=(8-x)(x+3)$

Step6: Factor $x^2-13x+42$

Find two numbers: $-6$ and $-7$ (sum $-13$, product $42$).
$x^2-13x+42=(x-6)(x-7)$

Step7: Factor $3x^2-5x+2$

Use AC method: $3*2=6$, find $-2$ and $-3$ (sum $-5$). Split and factor:
$3x^2-3x-2x+2=3x(x-1)-2(x-1)=(3x-2)(x-1)$

Step8: Factor $2x^2-x-1$

Use AC method: $2*(-1)=-2$, find $1$ and $-2$ (sum $-1$). Split and factor:
$2x^2+x-2x-1=x(2x+1)-1(2x+1)=(2x-1)(x+1)$

Step9: Factor $5x^2+26x+5$

Use AC method: $5*5=25$, find $25$ and $1$ (sum $26$). Split and factor:
$5x^2+25x+x+5=5x(x+5)+1(x+5)=(5x+1)(x+5)$

Step10: Factor $12x^2+7x+1$

Use AC method: $12*1=12$, find $3$ and $4$ (sum $7$). Split and factor:
$12x^2+3x+4x+1=3x(4x+1)+1(4x+1)=(3x+1)(4x+1)$

Step11: Factor $-9x^2+3x+2$

Rearrange and factor: $-(9x^2-3x-2)$. Use AC method: $9*(-2)=-18$, find $3$ and $-6$ (sum $-3$). Split and factor:
$9x^2+3x-6x-2=3x(3x+1)-2(3x+1)$, so $-9x^2+3x+2=-(3x-2)(3x+1)$

Step12: Factor $9x^2-4y^2$

Difference of squares: $a^2-b^2=(a-b)(a+b)$, where $a=3x$, $b=2y$.
$9x^2-4y^2=(3x-2y)(3x+2y)$

Step13: Factor $49-9x^2$

Difference of squares: $a^2-b^2=(a-b)(a+b)$, where $a=7$, $b=3x$.
$49-9x^2=(7-3x)(7+3x)$

Step14: Factor $x^2+49$

Sum of squares, cannot be factored over real numbers.
$x^2+49$ (prime over reals)

Step15: Factor $x^3-64$

Difference of cubes: $a^3-b^3=(a-b)(a^2+ab+b^2)$, where $a=x$, $b=4$.
$x^3-64=(x-4)(x^2+4x+16)$

Step16: Factor $5x^3+40$

Factor out GCF first: $5(x^3+8)$. Sum of cubes: $a^3+b^3=(a+b)(a^2-ab+b^2)$, where $a=x$, $b=2$.
$5x^3+40=5(x+2)(x^2-2x+4)$

Answer:

1. $(x+2)(x-1)$
2. $(x-6)(x+1)$
3. $(x-2)(x-3)$
4. $(x-3y)(x+2y)$
5. $(8-x)(x+3)$
6. $(x-6)(x-7)$
7. $(3x-2)(x-1)$
8. $(2x-1)(x+1)$
9. $(5x+1)(x+5)$
10. $(3x+1)(4x+1)$
11. $-(3x-2)(3x+1)$
12. $(3x-2y)(3x+2y)$
13. $(7-3x)(7+3x)$
14. $x^2+49$ (prime over real numbers)
15. $(x-4)(x^2+4x+16)$
16. $5(x+2)(x^2-2x+4)$