Factor the following polynomials by GCF and/or grouping. Show ALL work! $$ \begin{array}{ll} ext { 1.) } 5 x^{3}-6 x^{2}-5 x+6 & ext { 16.) } 3 x^{2}-27 \ ext { 2.) } 6 x^{3}+12 x^{2}+6 x & ext { 17.) } 3 x^{3}+3 x^{2}+9 x+9 \ ext { 3.) } x^{6}-x^{5}+x^{4}-x^{3} & ext { 18.) }-3 x^{6}+24 x^{3} \ ext { 4.) } 10 x^{3}+15 x^{2}+20 x+30 & ext { 19.) }-3 x^{6}-12 x^{3} \ ext { 5.) } 100 x^{7}-25 x^{5} & ext { 20.) } 30 x^{3}+25 x^{2}+42 x+35\end{array} $$

Question

Factor the following polynomials by GCF and/or grouping. Show ALL work! $$ \begin{array}{ll}	ext { 1.) } 5 x^{3}-6 x^{2}-5 x+6 & 	ext { 16.) } 3 x^{2}-27 \ 	ext { 2.) } 6 x^{3}+12 x^{2}+6 x & 	ext { 17.) } 3 x^{3}+3 x^{2}+9 x+9 \ 	ext { 3.) } x^{6}-x^{5}+x^{4}-x^{3} & 	ext { 18.) }-3 x^{6}+24 x^{3} \ 	ext { 4.) } 10 x^{3}+15 x^{2}+20 x+30 & 	ext { 19.) }-3 x^{6}-12 x^{3} \ 	ext { 5.) } 100 x^{7}-25 x^{5} & 	ext { 20.) } 30 x^{3}+25 x^{2}+42 x+35\end{array} $$

UpStudy AI · Accepted Answer

Factor the expression by following steps:
- step0: Factor:
  $$6x^{3}+12x^{2}+6x$$
- step1: Rewrite the expression:
  $$6x	imes x^{2}+6x	imes 2x+6x$$
- step2: Factor the expression:
  $$6x\left(x^{2}+2x+1ight)$$
- step3: Factor the expression:
  $$6x\left(x+1ight)^{2}$$
Factor the expression $$ 30 x^{3}+25 x^{2}+42 x+35 $$.
Factor the expression by following steps:
- step0: Factor:
  $$30x^{3}+25x^{2}+42x+35$$
- step1: Calculate:
  $$30x^{3}+42x+25x^{2}+35$$
- step2: Rewrite the expression:
  $$6x	imes 5x^{2}+6x	imes 7+5	imes 5x^{2}+5	imes 7$$
- step3: Factor the expression:
  $$6x\left(5x^{2}+7ight)+5\left(5x^{2}+7ight)$$
- step4: Factor the expression:
  $$\left(6x+5ight)\left(5x^{2}+7ight)$$
Factor the expression $$ 100 x^{7}-25 x^{5} $$.
Factor the expression by following steps:
- step0: Factor:
  $$100x^{7}-25x^{5}$$
- step1: Factor the expression:
  $$25x^{5}\left(4x^{2}-1ight)$$
- step2: Factor the expression:
  $$25x^{5}\left(2x-1ight)\left(2x+1ight)$$
Factor the expression $$ 3 x^{2}-27 $$.
Factor the expression by following steps:
- step0: Factor:
  $$3x^{2}-27$$
- step1: Factor the expression:
  $$3\left(x^{2}-9ight)$$
- step2: Factor the expression:
  $$3\left(x-3ight)\left(x+3ight)$$
Factor the expression $$ 3 x^{3}+3 x^{2}+9 x+9 $$.
Factor the expression by following steps:
- step0: Factor:
  $$3x^{3}+3x^{2}+9x+9$$
- step1: Evaluate:
  $$3x^{3}+9x+3x^{2}+9$$
- step2: Rewrite the expression:
  $$3x^{3}+3	imes 3x+3x^{2}+3	imes 3$$
- step3: Factor the expression:
  $$3\left(x^{3}+3x+x^{2}+3ight)$$
- step4: Factor the expression:
  $$3\left(x+1ight)\left(x^{2}+3ight)$$
Factor the expression $$ -3 x^{6}+24 x^{3 $$.
Factor the expression by following steps:
- step0: Factor:
  $$-3x^{6}+24x^{3}$$
- step1: Factor the expression:
  $$3x^{3}\left(-x^{3}+8ight)$$
- step2: Factor the expression:
  $$3x^{3}\left(2-xight)\left(4+2x+x^{2}ight)$$
Factor the expression $$ 5 x^{3}-6 x^{2}-5 x+6 $$.
Factor the expression by following steps:
- step0: Factor:
  $$5x^{3}-6x^{2}-5x+6$$
- step1: Rewrite the expression:
  $$x^{2}	imes 5x-x^{2}	imes 6-5x+6$$
- step2: Factor the expression:
  $$x^{2}\left(5x-6ight)-\left(5x-6ight)$$
- step3: Factor the expression:
  $$\left(x^{2}-1ight)\left(5x-6ight)$$
- step4: Factor the expression:
  $$\left(x-1ight)\left(x+1ight)\left(5x-6ight)$$
Factor the expression $$ x^{6}-x^{5}+x^{4}-x^{3 $$.
Factor the expression by following steps:
- step0: Factor:
  $$x^{6}-x^{5}+x^{4}-x^{3}$$
- step1: Evaluate:
  $$x^{6}+x^{4}-x^{5}-x^{3}$$
- step2: Rewrite the expression:
  $$x^{3}	imes x^{3}+x^{3}	imes x-x^{3}	imes x^{2}-x^{3}$$
- step3: Factor the expression:
  $$x^{3}\left(x^{3}+x-x^{2}-1ight)$$
- step4: Factor the expression:
  $$x^{3}\left(x-1ight)\left(x^{2}+1ight)$$
Factor the expression $$ 10 x^{3}+15 x^{2}+20 x+30 $$.
Factor the expression by following steps:
- step0: Factor:
  $$10x^{3}+15x^{2}+20x+30$$
- step1: Evaluate:
  $$10x^{3}+20x+15x^{2}+30$$
- step2: Rewrite the expression:
  $$5	imes 2x^{3}+5	imes 4x+5	imes 3x^{2}+5	imes 6$$
- step3: Factor the expression:
  $$5\left(2x^{3}+4x+3x^{2}+6ight)$$
- step4: Factor the expression:
  $$5\left(2x+3ight)\left(x^{2}+2ight)$$
Factor the expression $$ -3 x^{6}-12 x^{3 $$.
Factor the expression by following steps:
- step0: Factor:
  $$-3x^{6}-12x^{3}$$
- step1: Rewrite the expression:
  $$-3x^{3}	imes x^{3}-3x^{3}	imes 4$$
- step2: Factor the expression:
  $$-3x^{3}\left(x^{3}+4ight)$$
Here are the factorizations for the given polynomials, along with the steps involved in each case:

1. **Factor $$ 5x^{3} - 6x^{2} - 5x + 6 $$**:
   - Group the terms: $$ (5x^{3} - 6x^{2}) + (-5x + 6) $$
   - Factor out the GCF from each group:
     - From the first group: $$ x^{2}(5x - 6) $$
     - From the second group: $$ -1(5x - 6) $$
   - Combine: $$ (5x - 6)(x^{2} - 1) $$
   - Factor $$ x^{2} - 1 $$ further: $$ (x - 1)(x + 1) $$
   - Final factorization: **$$ (x - 1)(x + 1)(5x - 6) $$**

2. **Factor $$ 6x^{3} + 12x^{2} + 6x $$**:
   - Factor out the GCF: $$ 6x(x^{2} + 2x + 1) $$
   - Recognize $$ x^{2} + 2x + 1 $$ as a perfect square: $$ (x + 1)^{2} $$
   - Final factorization: **$$ 6x(x + 1)^{2} $$**

3. **Factor $$ x^{6} - x^{5} + x^{4} - x^{3} $$**:
   - Group the terms: $$ (x^{6} - x^{5}) + (x^{4} - x^{3}) $$
   - Factor out the GCF from each group:
     - From the first group: $$ x^{5}(x - 1) $$
     - From the second group: $$ x^{3}(x - 1) $$
   - Combine: $$ (x - 1)(x^{5} + x^{3}) $$
   - Factor $$ x^{5} + x^{3} $$: $$ x^{3}(x^{2} + 1) $$
   - Final factorization: **$$ x^{3}(x - 1)(x^{2} + 1) $$**

4. **Factor $$ 10x^{3} + 15x^{2} + 20x + 30 $$**:
   - Group the terms: $$ (10x^{3} + 15x^{2}) + (20x + 30) $$
   - Factor out the GCF from each group:
     - From the first group: $$ 5x^{2}(2x + 3) $$
     - From the second group: $$ 10(2x + 3) $$
   - Combine: $$ (2x + 3)(5x^{2} + 10) $$
   - Factor $$ 5x^{2} + 10 $$: $$ 5(x^{2} + 2) $$
   - Final factorization: **$$ 5(2x + 3)(x^{2} + 2) $$**

5. **Factor $$ 100x^{7} - 25x^{5} $$**:
   - Factor out the GCF: $$ 25x^{5}(4x^{2} - 1) $$
   - Recognize $$ 4x^{2} - 1 $$ as a difference of squares: $$ (2x - 1)(2x + 1) $$
   - Final factorization: **$$ 25x^{5}(2x - 1)(2x + 1) $$**

6. **Factor $$ 3x^{2} - 27 $$**:
   - Recognize $$ 3 $$ as a common factor: $$ 3(x^{2} - 9) $$
   - Factor $$ x^{2} - 9 $$ as a difference of squares: $$ (x - 3)(x + 3) $$
   - Final factorization: **$$ 3(x - 3)(x + 3) $$**

7. **Factor $$ 3x^{3} + 3x^{2} + 9x + 9 $$**:
   - Group the terms: $$ (3x^{3} + 3x^{2}) + (9x + 9) $$
   - Factor out the GCF from each group:
     - From the first group: $$ 3x^{2}(x + 1) $$
     - From the second group: $$ 9(x + 1) $$
   - Combine: $$ (x + 1)(3x^{2} + 9) $$
   - Factor $$ 3x^{2} + 9 $$: $$ 3(x^{2} + 3) $$
   - Final factorization: **$$ 3(x + 1)(x^{2} + 3) $$**

8. **Factor $$ -3x^{6} + 24x^{3} $$**:
   - Factor out the GCF: $$ -3x^{3}(x^{3} - 8) $$
   - Recognize $$ x^{3} - 8 $$ as a difference of cubes: $$ (x - 2)(x^{2} + 2x + 4) $$
   - Final factorization: **$$ -3x^{3}(x - 2)(x^{2} + 2x + 4) $$**

9. **Factor $$ -3x^{6} - 12x^{3} $$**:
   - Factor out the GCF: $$ -3x^{3}(x^{3} + 4) $$
   - Final factorization: **$$ -3x^{3}(x^{3} + 4) $$**

10. **Factor $$ 30x^{3} + 25x^{2} + 42x + 35 $$**:
    - Group the terms: $$ (30x^{3} + 25x^{2}) + (42x + 35) $$
    - Factor out the GCF from each group:
      - From the first group: $$ 5x^{2}(6x + 5) $$
      - From the second group: $$ 7(6x + 5) $$
    - Combine: $$ (6x + 5)(5x^{2} + 7) $$
    - Final factorization: **$$ (6x + 5)(5x^{2} + 7) $$**

These are the complete factorizations for the given polynomials.
Here are the factorizations for the polynomials: 1. $$ 5x^{3} - 6x^{2} - 5x + 6 = (x - 1)(x + 1)(5x - 6) $$ 2. $$ 6x^{3} + 12x^{2} + 6x = 6x(x + 1)^{2} $$ 3. $$ x^{6} - x^{5} + x^{4} - x^{3} = x^{3}(x - 1)(x^{2} + 1) $$ 4. $$ 10x^{3} + 15x^{2} + 20x + 30 = 5(2x + 3)(x^{2} + 2) $$ 5. $$ 100x^{7} - 25x^{5} = 25x^{5}(2x - 1)(2x + 1) $$ 6. $$ 3x^{2} - 27 = 3(x - 3)(x + 3) $$ 7. $$ 3x^{3} + 3x^{2} + 9x + 9 = 3(x + 1)(x^{2} + 3) $$ 8. $$ -3x^{6} + 24x^{3} = -3x^{3}(x - 2)(x^{2} + 2x + 4) $$ 9. $$ -3x^{6} - 12x^{3} = -3x^{3}(x^{3} + 4) $$ 10. $$ 30x^{3} + 25x^{2} + 42x + 35 = (6x + 5)(5x^{2} + 7) $$

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