Question

1. $x^{2}-16$
2. $y^{2}-9$
3. $9x^{2}-25$
4. $4a^{2}-49$
5. $4x^{2}-25$
6. $100y^{2}-81$
7. $6x^{2}-6y^{2}$
8. $8x^{2}-8y^{2}$
9. $3x^{8}-3y^{8}$
10. $5x^{4}-5y^{4}$
11. $4xy^{4}-4xz^{4}$
12. $9a^{4}-a^{2}b^{2}$
13. $a^{2}+2ab+b^{2}-9$
14. $x^{2}-2xy+y^{2}-25$
15. $r^{2}-2r+1-4s^{2}$
16. $(c^{2}+4cd+4d^{2})-9p^{2}$
17. $2m^{2}+4mn+2n^{2}-50b^{2}$
18. $12x^{2}+12x+3-3y^{2}$
19. $9-(a^{2}+2ab+b^{2})$
20. $16-(x^{2}-2xy+y^{2})$
21. $ac+ad+bc+bd$
22. $xy+xz+wy+wz$
23. $b^{3}-b^{2}+2b-2$
24. $y^{3}-y^{2}+3y-3$
25. $y^{2}-8y-y+8$
26. $t^{2}+6t-2t-12$
27. $2y^{4}+6y^{2}+5y^{2}+15$
28. $2xy-x^{2}y-6+3x$

Explanation:

All problems involve factoring polynomials, primarily using difference of squares, perfect square trinomials, and factoring by grouping. Below are the step-by-step solutions for each:

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Problem 36: $x^2 - 16$

Step1: Recognize difference of squares

$x^2 - 4^2$

Step2: Apply $a^2-b^2=(a-b)(a+b)$

$(x-4)(x+4)$

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Problem 37: $y^2 - 9$

Step1: Recognize difference of squares

$y^2 - 3^2$

Step2: Apply difference of squares rule

$(y-3)(y+3)$

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Problem 38: $9x^2 - 25$

Step1: Recognize difference of squares

$(3x)^2 - 5^2$

Step2: Apply difference of squares rule

$(3x-5)(3x+5)$

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Problem 39: $4a^2 - 49$

Step1: Recognize difference of squares

$(2a)^2 - 7^2$

Step2: Apply difference of squares rule

$(2a-7)(2a+7)$

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Problem 40: $4x^2 - 25$

Step1: Recognize difference of squares

$(2x)^2 - 5^2$

Step2: Apply difference of squares rule

$(2x-5)(2x+5)$

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Problem 41: $100y^2 - 81$

Step1: Recognize difference of squares

$(10y)^2 - 9^2$

Step2: Apply difference of squares rule

$(10y-9)(10y+9)$

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Problem 42: $6x^2 - 6y^2$

Step1: Factor out GCF 6

$6(x^2 - y^2)$

Step2: Factor difference of squares

$6(x-y)(x+y)$

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Problem 43: $8x^2 - 8y^2$

Step1: Factor out GCF 8

$8(x^2 - y^2)$

Step2: Factor difference of squares

$8(x-y)(x+y)$

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Problem 44: $3x^8 - 3y^8$

Step1: Factor out GCF 3

$3(x^8 - y^8)$

Step2: Factor difference of squares

$3[(x^4)^2 - (y^4)^2] = 3(x^4-y^4)(x^4+y^4)$

Step3: Factor $x^4-y^4$ again

$3[(x^2)^2-(y^2)^2](x^4+y^4) = 3(x^2-y^2)(x^2+y^2)(x^4+y^4)$

Step4: Factor $x^2-y^2$ again

$3(x-y)(x+y)(x^2+y^2)(x^4+y^4)$

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Problem 45: $5x^4 - 5y^4$

Step1: Factor out GCF 5

$5(x^4 - y^4)$

Step2: Factor difference of squares

$5[(x^2)^2-(y^2)^2] = 5(x^2-y^2)(x^2+y^2)$

Step3: Factor $x^2-y^2$ again

$5(x-y)(x+y)(x^2+y^2)$

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Problem 46: $4xy^4 - 4xz^4$

Step1: Factor out GCF $4x$

$4x(y^4 - z^4)$

Step2: Factor difference of squares

$4x[(y^2)^2-(z^2)^2] = 4x(y^2-z^2)(y^2+z^2)$

Step3: Factor $y^2-z^2$ again

$4x(y-z)(y+z)(y^2+z^2)$

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Problem 47: $9a^4 - a^2b^2$

Step1: Factor out GCF $a^2$

$a^2(9a^2 - b^2)$

Step2: Factor difference of squares

$a^2[(3a)^2 - b^2] = a^2(3a-b)(3a+b)$

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Problem 48: $a^2 + 2ab + b^2 - 9$

Step1: Factor perfect square trinomial

$(a+b)^2 - 3^2$

Step2: Factor difference of squares

$(a+b-3)(a+b+3)$

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Problem 49: $x^2 - 2xy + y^2 - 25$

Step1: Factor perfect square trinomial

$(x-y)^2 - 5^2$

Step2: Factor difference of squares

$(x-y-5)(x-y+5)$

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Problem 50: $r^2 - 2r + 1 - 4s^2$

Step1: Factor perfect square trinomial

$(r-1)^2 - (2s)^2$

Step2: Factor difference of squares

$(r-1-2s)(r-1+2s)$

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Problem 51: $(c^2 + 4cd + 4d^2) - 9p^2$

Step1: Factor perfect square trinomial

$(c+2d)^2 - (3p)^2$

Step2: Factor difference of squares

$(c+2d-3p)(c+2d+3p)$

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Problem 52: $2m^2 + 4mn + 2n^2 - 50b^2$

Step1: Factor out GCF 2

$2(m^2 + 2mn + n^2 - 25b^2)$

Step2: Factor perfect square trinomial

$2[(m+n)^2 - (5b)^2]$

Step3: Factor difference of squares

$2(m+n-5b)(m+n+5b)$

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Problem 53: $12x^2 + 12x + 3 - 3y^2$

Step1: Factor out GCF 3

$3(4x^2 + 4x + 1 - y^2)$

Step2: Factor perfect square trinomial

$3[(2x+1)^2 - y^2]$

Step3: Factor difference of squares

$3(2x+1-y)(2x+1+y)$

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Problem 54: $9 - (a^2 + 2ab + b^2)$

Step1: Factor perfect square trinomial

$3^2 - (a+b)^2$

Step2: Factor difference of squares

$(3-a-b)(3+a+b)$

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Problem 55: $16 - (x^2 - 2xy + y^2)$

Step1: Factor perfect square trinomial

$4^2 - (x-y)^2$

Step2: Factor difference of squares

$(4-x+y)(4+x-y)$

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Problem 56: $ac + ad + bc + bd$

Step1: Group terms

$(ac+ad)+(bc+bd)$

Step2: Factor GCF from each group

$a(c+d)+b(c+d)$

Step3: Factor out common binomial

$(a+b)(c+d)$

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Problem 57: $xy + xz + wy + wz$

Step1: Group terms

$(xy+xz)+(wy+wz)$

Step2: Factor GCF from each group

$x(y+z)+w(y+z)$

Step3: Factor out common binomial

$(x+w)(y+z)$

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Problem 58: $b^3 - b^2 + 2b - 2$

Step1: Group terms

$(b^3-b^2)+(2b-2)$

Step2: Factor GCF from each group

$b^2(b-1)+2(b-1)$

Step3: Factor out common binomial

$(b^2+2)(b-1)$

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Problem 59: $y^3 - y^2 + 3y - 3$

Step1: Group terms

$(y^3-y^2)+(3y-3)$

Step2: Factor GCF from each group

$y^2(y-1)+3(y-1)$

Step3: Factor out common binomial

$(y^2+3)(y-1)$

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Problem 60: $y^2 - 8y - y + 8$

Step1: Group terms

$(y^2-8y)+(-y+8)$

Step2: Factor GCF from each group

$y(y-8)-1(y-8)$

Step3: Factor out common binomial

$(y-1)(y-8)$

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Problem 61: $t^2 + 6t - 2t - 12$

Step1: Group terms

$(t^2+6t)+(-2t-12)$

Step2: Factor GCF from each group

$t(t+6)-2(t+6)$

Step3: Factor out common binomial

$(t-2)(t+6)$

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Problem 62: $2y^4 + 6y^2 + 5y^2 + 15$

Step1: Combine like terms

$2y^4 + 11y^2 + 15$

Step2: Factor quadratic in $y^2$

$(2y^2+5)(y^2+3)$

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Problem 63: $2xy - x^2y - 6 + 3x$

Step1: Rearrange and group terms

$(2xy-x^2y)+(-6+3x)$

Step2: Factor GCF from each group

$xy(2-x)+3(-2+x) = xy(2-x)-3(2-x)$

Step3: Factor out common binomial

$(xy-3)(2-x)$

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Answer:

1. $(x-4)(x+4)$
2. $(y-3)(y+3)$
3. $(3x-5)(3x+5)$
4. $(2a-7)(2a+7)$
5. $(2x-5)(2x+5)$
6. $(10y-9)(10y+9)$
7. $6(x-y)(x+y)$
8. $8(x-y)(x+y)$
9. $3(x-y)(x+y)(x^2+y^2)(x^4+y^4)$
10. $5(x-y)(x+y)(x^2+y^2)$
11. $4x(y-z)(y+z)(y^2+z^2)$
12. $a^2(3a-b)(3a+b)$
13. $(a+b-3)(a+b+3)$
14. $(x-y-5)(x-y+5)$
15. $(r-1-2s)(r-1+2s)$
16. $(c+2d-3p)(c+2d+3p)$
17. $2(m+n-5b)(m+n+5b)$
18. $3(2x+1-y)(2x+1+y)$
19. $(3-a-b)(3+a+b)$
20. $(4-x+y)(4+x-y)$
21. $(a+b)(c+d)$
22. $(x+w)(y+z)$
23. $(b^2+2)(b-1)$
24. $(y^2+3)(y-1)$
25. $(y-1)(y-8)$
26. $(t-2)(t+6)$
27. $(2y^2+5)(y^2+3)$
28. $(xy-3)(2-x)$