EXERCISE 4 Fully simplify each of the following expres 1. $$ (2 x+y)\left(3 x^{2}+2 x y+5 y^{2}\right) $$ 3. $$ (x-y)(3 x-2 x y+4) $$ 5. $$ (7 m-3 n)\left(5 m^{2}-2 m m-7 n^{2}\right) $$ 7. $$ (2 x-y)\left(4 x^{2}+2 x y-y^{2}\right) $$ 9. $$ (4 m-n)\left(2 m^{2}-3 m m-n^{2}\right) $$ 11. $$ (2 x-y)(2 x+y)(3 x-5 x y-y) $$ 13. $$ (9 m-n)\left(3 m^{2}-m m+2 n^{2}\right) $$ 15. $$ 5(a-b)\left(a^{2}+a b+b^{2}\right) $$ 17. $$ 2(3 p+q)\left(9 p^{2}-3 p q+q^{2}\right) $$ 19. $$ 3(2 x+1)\left(x^{2}-5 x-1\right) $$ 1

Question

UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Calculate:
  $$5\left(a-bight)\left(a^{2}+ab+b^{2}ight)$$
- step1: Simplify the product:
  $$5\left(a^{3}-b^{3}ight)$$
- step2: Calculate:
  $$5a^{3}-5b^{3}$$
Expand the expression $$ 2(3 p+q)(9 p^{2}-3 p q+q^{2}) $$
Simplify the expression by following steps:
- step0: Calculate:
  $$2\left(3p+qight)\left(9p^{2}-3pq+q^{2}ight)$$
- step1: Simplify the product:
  $$2\left(27p^{3}+q^{3}ight)$$
- step2: Calculate:
  $$54p^{3}+2q^{3}$$
Expand the expression $$ (4 m-n)(2 m^{2}-3 m m-n^{2}) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(4m-night)\left(2m^{2}-3m	imes m-n^{2}ight)$$
- step1: Multiply the terms:
  $$\left(4m-night)\left(2m^{2}-3m^{2}-n^{2}ight)$$
- step2: Subtract the terms:
  $$\left(4m-night)\left(-m^{2}-n^{2}ight)$$
- step3: Apply the distributive property:
  $$4m\left(-m^{2}ight)-4mn^{2}-n\left(-m^{2}ight)-\left(-n	imes n^{2}ight)$$
- step4: Multiply the terms:
  $$-4m^{3}-4mn^{2}+nm^{2}-\left(-n^{3}ight)$$
- step5: Remove the parentheses:
  $$-4m^{3}-4mn^{2}+nm^{2}+n^{3}$$
Expand the expression $$ (9 m-n)(3 m^{2}-m m+2 n^{2}) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(9m-night)\left(3m^{2}-m	imes m+2n^{2}ight)$$
- step1: Multiply the terms:
  $$\left(9m-night)\left(3m^{2}-m^{2}+2n^{2}ight)$$
- step2: Subtract the terms:
  $$\left(9m-night)\left(2m^{2}+2n^{2}ight)$$
- step3: Apply the distributive property:
  $$9m	imes 2m^{2}+9m	imes 2n^{2}-n	imes 2m^{2}-n	imes 2n^{2}$$
- step4: Multiply the terms:
  $$18m^{3}+18mn^{2}-2nm^{2}-2n^{3}$$
Expand the expression $$ (7 m-3 n)(5 m^{2}-2 m m-7 n^{2}) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(7m-3night)\left(5m^{2}-2m	imes m-7n^{2}ight)$$
- step1: Multiply the terms:
  $$\left(7m-3night)\left(5m^{2}-2m^{2}-7n^{2}ight)$$
- step2: Subtract the terms:
  $$\left(7m-3night)\left(3m^{2}-7n^{2}ight)$$
- step3: Apply the distributive property:
  $$7m	imes 3m^{2}-7m	imes 7n^{2}-3n	imes 3m^{2}-\left(-3n	imes 7n^{2}ight)$$
- step4: Multiply the terms:
  $$21m^{3}-49mn^{2}-9nm^{2}-\left(-21n^{3}ight)$$
- step5: Remove the parentheses:
  $$21m^{3}-49mn^{2}-9nm^{2}+21n^{3}$$
Expand the expression $$ 3(2 x+1)(x^{2}-5 x-1) $$
Simplify the expression by following steps:
- step0: Calculate:
  $$3\left(2x+1ight)\left(x^{2}-5x-1ight)$$
- step1: Multiply the terms:
  $$\left(6x+3ight)\left(x^{2}-5x-1ight)$$
- step2: Apply the distributive property:
  $$6x	imes x^{2}-6x	imes 5x-6x	imes 1+3x^{2}-3	imes 5x-3	imes 1$$
- step3: Multiply the terms:
  $$6x^{3}-30x^{2}-6x+3x^{2}-15x-3$$
- step4: Add the terms:
  $$6x^{3}-27x^{2}-21x-3$$
Expand the expression $$ (x-y)(3 x-2 x y+4) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(x-yight)\left(3x-2xy+4ight)$$
- step1: Apply the distributive property:
  $$x	imes 3x-x	imes 2xy+x	imes 4-y	imes 3x-\left(-y	imes 2xyight)-y	imes 4$$
- step2: Multiply the terms:
  $$3x^{2}-2x^{2}y+4x-3yx-\left(-2y^{2}xight)-4y$$
- step3: Remove the parentheses:
  $$3x^{2}-2x^{2}y+4x-3yx+2y^{2}x-4y$$
Expand the expression $$ (2 x-y)(4 x^{2}+2 x y-y^{2}) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(2x-yight)\left(4x^{2}+2xy-y^{2}ight)$$
- step1: Apply the distributive property:
  $$2x	imes 4x^{2}+2x	imes 2xy-2xy^{2}-y	imes 4x^{2}-y	imes 2xy-\left(-y	imes y^{2}ight)$$
- step2: Multiply the terms:
  $$8x^{3}+4x^{2}y-2xy^{2}-4yx^{2}-2y^{2}x-\left(-y^{3}ight)$$
- step3: Remove the parentheses:
  $$8x^{3}+4x^{2}y-2xy^{2}-4yx^{2}-2y^{2}x+y^{3}$$
- step4: Subtract the terms:
  $$8x^{3}+0-2xy^{2}-2y^{2}x+y^{3}$$
- step5: Remove 0:
  $$8x^{3}-2xy^{2}-2y^{2}x+y^{3}$$
- step6: Subtract the terms:
  $$8x^{3}-4xy^{2}+y^{3}$$
Expand the expression $$ (2 x+y)(3 x^{2}+2 x y+5 y^{2}) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(2x+yight)\left(3x^{2}+2xy+5y^{2}ight)$$
- step1: Apply the distributive property:
  $$2x	imes 3x^{2}+2x	imes 2xy+2x	imes 5y^{2}+y	imes 3x^{2}+y	imes 2xy+y	imes 5y^{2}$$
- step2: Multiply the terms:
  $$6x^{3}+4x^{2}y+10xy^{2}+3yx^{2}+2y^{2}x+5y^{3}$$
- step3: Add the terms:
  $$6x^{3}+7x^{2}y+12xy^{2}+5y^{3}$$
Expand the expression $$ (2 x-y)(2 x+y)(3 x-5 x y-y) $$
Simplify the expression by following steps:
- step0: Calculate:
  $$\left(2x-yight)\left(2x+yight)\left(3x-5xy-yight)$$
- step1: Multiply the terms:
  $$\left(4x^{2}-y^{2}ight)\left(3x-5xy-yight)$$
- step2: Apply the distributive property:
  $$4x^{2}	imes 3x-4x^{2}	imes 5xy-4x^{2}y-y^{2}	imes 3x-\left(-y^{2}	imes 5xyight)-\left(-y^{2}	imes yight)$$
- step3: Multiply the terms:
  $$12x^{3}-20x^{3}y-4x^{2}y-3y^{2}x-\left(-5y^{3}xight)-\left(-y^{3}ight)$$
- step4: Remove the parentheses:
  $$12x^{3}-20x^{3}y-4x^{2}y-3y^{2}x+5y^{3}x+y^{3}$$
Here are the fully simplified expressions for each of the given problems:

1. **Expression:** $$ (2 x+y)(3 x^{2}+2 x y+5 y^{2}) $$  
   **Simplified Result:** $$ 6x^{3}+7x^{2}y+12xy^{2}+5y^{3} $$

2. **Expression:** $$ (x-y)(3 x-2 x y+4) $$  
   **Simplified Result:** $$ 3x^{2}-2x^{2}y+4x-3yx+2y^{2}x-4y $$

3. **Expression:** $$ (7 m-3 n)(5 m^{2}-2 m m-7 n^{2}) $$  
   **Simplified Result:** $$ 21m^{3}-49mn^{2}-9nm^{2}+21n^{3} $$

4. **Expression:** $$ (2 x-y)(4 x^{2}+2 x y-y^{2}) $$  
   **Simplified Result:** $$ 8x^{3}-4xy^{2}+y^{3} $$

5. **Expression:** $$ (4 m-n)(2 m^{2}-3 m m-n^{2}) $$  
   **Simplified Result:** $$ -4m^{3}-4mn^{2}+nm^{2}+n^{3} $$

6. **Expression:** $$ (2 x-y)(2 x+y)(3 x-5 x y-y) $$  
   **Simplified Result:** $$ 12x^{3}-20x^{3}y-4x^{2}y-3y^{2}x+5y^{3}x+y^{3} $$

7. **Expression:** $$ (9 m-n)(3 m^{2}-m m+2 n^{2}) $$  
   **Simplified Result:** $$ 18m^{3}+18mn^{2}-2nm^{2}-2n^{3} $$

8. **Expression:** $$ 5(a-b)(a^{2}+a b+b^{2}) $$  
   **Simplified Result:** $$ 5a^{3}-5b^{3} $$

9. **Expression:** $$ 2(3 p+q)(9 p^{2}-3 p q+q^{2}) $$  
   **Simplified Result:** $$ 54p^{3}+2q^{3} $$

10. **Expression:** $$ 3(2 x+1)(x^{2}-5 x-1) $$  
    **Simplified Result:** $$ 6x^{3}-27x^{2}-21x-3 $$

If you need further assistance or explanations for any specific expression, feel free to ask!
Here are the fully simplified expressions for each problem: 1. $$ (2x + y)(3x^2 + 2xy + 5y^2) = 6x^3 + 7x^2y + 12xy^2 + 5y^3 $$ 2. $$ (x - y)(3x - 2xy + 4) = 3x^2 - 2x^2y + 4x - 3xy + 2y^2x - 4y $$ 3. $$ (7m - 3n)(5m^2 - 2mm - 7n^2) = 21m^3 - 49mn^2 - 9nm^2 + 21n^3 $$ 4. $$ (2x - y)(4x^2 + 2xy - y^2) = 8x^3 - 4xy^2 + y^3 $$ 5. $$ (4m - n)(2m^2 - 3mm - n^2) = -4m^3 - 4mn^2 + nm^2 + n^3 $$ 6. $$ (2x - y)(2x + y)(3x - 5xy - y) = 12x^3 - 20x^3y - 4x^2y - 3y^2x + 5y^3x + y^3 $$ 7. $$ (9m - n)(3m^2 - mm + 2n^2) = 18m^3 + 18mn^2 - 2nm^2 - 2n^3 $$ 8. $$ 5(a - b)(a^2 + ab + b^2) = 5a^3 - 5b^3 $$ 9. $$ 2(3p + q)(9p^2 - 3pq + q^2) = 54p^3 + 2q^3 $$ 10. $$ 3(2x + 1)(x^2 - 5x - 1) = 6x^3 - 27x^2 - 21x - 3 $$ If you need more help with any of these, let me know!

EXERCISE 4
Fully simplify each of the following expres

1

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