**Exercise 6.5**

**Multiply:**

**1. (5x + 3) by (7x + 2)**

**Solution: **

To multiply, we will use distributive law as follows:

(5x+3)(7x+2)

=5x(7x+2)+3(7x+2)

=(5x×7x+5x×2)+(3×7x+3×2)

=(35x^{2}+10x)+(21x+6)

=35x^{2}+10x+21x+6

=35x^{2}+31x+6

Thus, the answer is 35×2+31x+6

**2. (2x + 8) by (x – 3)**

**Solution: **

To multiply, we will use distributive law as follows:

(2x+8) (x–3)

= 2x(x–3) +8 (x–3)

= (2x × x–2x × 3) + (8x–8×3)

= (2x^{2}–6x) + (8x–24)

= 2x^{2}–6x + 8x–24 = 2x^{2}+2x–24

Thus, the answer is 2×2+2x–24.

**3. (7x + y) by (x + 5y)**

**Solution: **

To multiply, we will use distributive law as follows:

(7x+y)(x+5y)

=7x(x+5y)+y(x+5y)

=7x^{2}+35xy+xy+5y^{2}

=7x^{2}+36xy+5y^{2}

Thus, the answer is 7x^{2}+36xy+5y^{2}.

**4. (a – 1) by (0.1a2 + 3)**

**Solution: **

To multiply, we will use distributive law as follows:

(a–1)(0.1a^{2}+3)

=0.1a^{2}(a–1)+3(a–1)

=0.1a^{3}–0.1a^{2}+3a–3

Thus, the answer is 0.1a^{3}–0.1a^{2}+3a–3.

**5. (3x**^{2}+y^{2})(2x^{2}+3y^{2})

^{2}+y

^{2})(2x

^{2}+3y

^{2})

**Solution: **

To multiply, we will use distributive law as follows:

(3x^{2}+y^{2})(2x^{2}+3y^{2})

=3x^{2}(2x^{2}+3y^{2})+y^{2}(2x^{2}+3y^{2})

=6x^{4}+9x^{2}y^{2}+2x^{2}y^{2}+3y^{4}

=6x^{4}+11x^{2}y^{2}+3y^{4}

Thus, the answer is 6x^{4}+11x^{2}y2+3y^{4}.

**7. (****x****6****–****y****6****) (****x****2****+****y****2****)**

** ****Solution **

To multiply, we will use distributive law as follows:

(x^{6}–y^{6}) (x^{2}+y^{2})

=x^{6}(x^{2}+y^{2}) – y6(x^{2}+y^{2})

=(x^{8}+x^{6}y2) – (y^{6}x^{2}+y^{8})

=x^{8}+x^{6}y^{2}–y^{6}x^{2}–y^{8}

Thus, the answer is x^{8}+x^{6}y^{2}–y^{6}x^{2}–y^{8 }

**8. (x**^{2}+y^{2}) (3a+2b)

^{2}+y

^{2}) (3a+2b)

**Solution:**

To multiply, we will use distributive law as follows:

(x^{2}+y^{2}) (3a+2b)

=x^{2}(3a+2b)+y^{2}(3a+2b)

=3ax^{2}+2bx^{2}+3ay^{2}+2by^{2}

Thus, the answer is 3ax^{2}+2bx^{2}+3ay^{2}+2by^{2}

**9. [–****3****d****+(–****7****f****)] (****5****d****+****f****)**

**Solution:**

To multiply, we will use distributive law as follows:

[–3d+(–7f)](5d+f)

= (–3d)(5d+f) + (–7f)(5d+f)

= (–15d^{2}–3df) + (–35df–7f^{2})

= –15d^{2}–3df–35df–7f^{2}

= –15d^{2}–38df–7f^{2}

Thus, the answer is −15d^{2}–38df–7f^{2}.

**10. (****0.8****a****–****0.5****b****) (****1.5****a****–****3****b****)**

**Solution: **

To multiply, we will use distributive law as follows:

(0.8a–0.5b) (1.5a–3b)

= 0.8a (1.5a–3b) – 0.5b (1.5a–3b)

= 1.2a^{2}–2.4ab –0.75ab+1.5b^{2}

= 1.2a^{2}–3.15ab+1.5b^{2}

Thus, the answer is 1.2a2–3.15ab+1.5b2.

**11. (****2****x****2****y****2****–****5****xy****2****)(****x****2****–****y****2****)**

**Solution:**

To multiply, we will use distributive law as follows:

(2x2y^{2}–5xy^{2}) (x^{2}–y^{2})

= 2x^{2}y^{2} (x^{2}–y^{2})–5xy^{2} (x^{2}–y^{2})

= 2x^{4}y^{2}–2x^{2}y^{4}–5x^{3}y^{2}+5xy^{4}

Thus, the answer is 2x^{4}y^{2}–2x^{2}y^{4}–5x^{3}y^{2}+5xy^{4}.

** **

**14. (****3****x**^{2}**y****–****5****xy**^{2}**)(****15****x**^{2}**+****13****y**^{2}**)**

^{2}

^{2}

^{2}

^{2}

** ****Solution:**

To multiply, we will use distributive law as follows:

(3x^{2}y–5xy^{2})(15x^{2}+13y^{2})

=15x^{2}(3x^{2}y–5xy^{2})+13y^{2}(3x^{2}y–5xy^{2})

=35x^{4}y–x^{3}y^{2}+x^{2}y^{3}–5^{3}xy^{4}

Thus, the answer is 35x^{4}y–x^{3}y^{2}+x^{2}y^{3}–5^{3}xy^{4}.

**15. (****2****x****2****–****1****) (****4****x****3****+****5****x****2****)**

** ****Solution: **

To multiply, we will use distributive law as follows:

(2x^{2}–1) (4x^{3}+5x^{2})

=2x^{2}(4x^{3}+5x^{2})–1(4x^{3}+5x^{2})

=8x^{5}+10x^{4}–4x^{3}–5x^{2}

Thus, the answer is 8x^{5}+10x^{4}–4x^{3}–5x^{2}.

**16. (****2****xy****+****3****y****2****) (****3****y****2****–****2****)**

** ****Solution:**

To multiply, we will use distributive law as follows:

(2xy+3y2)(3y2–2)

=2xy(3y^{2}–2)+3y^{2}(3y^{2}–2)

=6xy^{3}–4xy+9y^{4}–6y^{2}

=9y^{4}+6xy^{3}–6y^{2}–4xy

Thus, the answer is 9y^{4}+6xy^{3}–6y^{2}–4xy.

**Find the following products and verify the result for x = -1 and y = -2:**

**17. (****3****x****–****5****y****) (****x****+****y****)**

**Solution: **

To multiply, we will use distributive law as follows:

(3x–5y)(x+y)

= 3x(x+y)–5y(x+y)

= 3x^{2}+3xy–5xy–5y^{2}

=3x^{2}–2xy –5y^{2}

= ∴(3x–5y) (x+y)

= 3x^{2 }–2xy –5y^{2}

Now, we put x = -1 and y = -2 on both sides to verify the result.

LHS = (3x–5y) (x+y)

= {3(−1) –5(−2)} {−1+ (−2)}

= (−3+10) (−3)

= −21

RHS = 3x^{2} – 2xy–5y^{2}

=3(−1)2–2(−1)(−2)–5(−2)2

=3×1–4–5×4

=3–4–20

=−21

Because LHS is equal to RHS, the result is verified.

**18. (****x****2****y****–****1****) (****3****–****2****x****2****y****)**

**Solution: **

To multiply, we will use distributive law as follows:

(x^{2}y–1)(3–2x^{2}y)

=x^{2}y(3–2x^{2}y)–1(3–2x^{2}y)

=3x^{2}y–2x^{4}y^{2}–3+22y

=5x^{2}y–2x^{4}y^{2}–3

∴ (x^{2}y–1)(3–2x^{2}y)=5x^{2}y–2x^{4}y^{2}–3

Now, we put x = -1 and y = -2 on both sides to verify the result.

LHS = (x^{2}y–1)(3–2x^{2}y)

= [(−1)2(−2)–1][3–2(−1)2(−2)]

= [1×(−2)–1][3–2×1×(−2)]

= (−2–1)(3+4)

= −3×7

= −21

RHS = 5x^{2}y–2x^{4}y^{2}–3

= 5(−1)2(−2)–2(−1)4(−2)2–3

= [5×1×(−2)]–[2×1×4]–3

= –10–8–3

=−21

Because LHS is equal to RHS, the result is verified.

**Simplify:**

** 20. ****x**^{2} **(****x****+****2****y****) (****x****–****3****y****)**

^{2}

** ****Solution:**

To simplify, we will use distributive law as follows:

x^{2 }(x+2y)(x–3y)

=[x^{2}(x+^{2}y)] (x–3y)

=(x^{3}+2x^{2}y)(x–3y)

=x3(x–3y) +2x^{2}y(x–3y)

=x^{4}–3x^{3}+2x^{3}–6x^{2}y^{2}

=x^{4}–x^{3}–6x^{2}y^{2}

Thus, the answer is x^{4}–x^{3}–6x^{2}y^{2}.

**21. (****x**^{2}**–****2****y**^{2}**)(****x****+****4****y****)****x****2****y****2**

^{2}

^{2}

** ****Solution: **

To simplify, we will use distributive law as follows:

(x^{2}–2y^{2})(x+4y)x^{2}y^{2}

=[x^{2}(x+4y)–2y^{2}(x+4y)]x^{2}y^{2}

=(x^{3}+4x^{2}y–2xy^{2}–8y^{3})x^{2}y^{2}

=x^{5}y^{2}+4x^{4}y^{3}–2x^{3}y^{4}–8x^{2}y^{5}

Thus, the answer is x^{5}y^{2}+4x^{4}y^{3}–2x^{3}y^{4}–8x^{2}y^{5}.

**22. a**^{2}**b**^{2 }**(****a****+****2****b****)(****3****a****+****b****)**

^{2}

^{2 }

**Solution: **

To simplify, we will use distributive law as follows:

a^{2}b^{2}(a+2b) (3a+b)

= [a^{2}b^{2}(a+2b)] (3a+b)

= (a^{3}b^{2}+2a^{2}b^{3}) (3a+b)

= 3a(a^{3}b^{2}+2a^{2}b^{3})+b(a^{3}b^{2}+2a^{2}b^{3})

= 3a^{4}b^{2}+6a^{3}b^{3}+a^{3}b^{3}+2a^{2}b^{4}

= 3a^{4}b^{2}+7a^{3}b^{3}+2a^{2}b^{4}

Thus, the answer is 3a^{4}b^{2}+7a^{3}b^{3}+2a^{2}b^{4}.

**23. x**^{2} **(****x****–****y****)****y**^{2} **(****x****+****2****y****)**

^{2}

^{2}

**Solution: **

To simplify, we will use distributive law as follows:

x^{2}(x–y) y^{2 }(x+2y)

= [x^{2}(x–y)] [y^{2}(x+2y)]

= (x^{3}–x^{2}y) (xy^{2}+2y^{3})

= x^{3}(xy^{2}+2y^{3})–x^{2}y(xy^{2}+2y^{3})

= x^{4}y^{2}+2x^{3}y^{3}–x^{3}y^{3}–2x^{2}y^{4}

= x4y^{2}+x^{3}y^{3}–2x^{2}y^{4}

Thus, the answer is x4y^{2}+x^{3}y^{3}–2x^{2}y^{4}

**24. (****x**^{3}**–****2****x**^{2}**+****5****x****–****7****)(****2****x****–****3****)**

^{3}

^{2}

**Solution: **

To simplify, we will use distributive law as follows:

(x^{3}–2x^{2}+5x–7)(2x–3)

=2x(x^{3}–2x^{2}+5x–7)–3(x^{3}–2x^{2}+5x–7)

=2x^{4}–4x^{3}+10x^{2}–14x–3x^{3}+6x^{2}–15x+21

=2x^{4}–4x^{3}–3x^{3}+10x^{2}+6x^{2}–14x–15x+21

=2x^{4}–7x^{3}+16x^{2}–29x+21

Thus, the answer is 2×4–7×3+16×2–29x+21.

**25. 2****x****4****–****7****x****3****+****16****x****2****–****29****x****+****21**

**Solution: **

To simplify, we will use distributive law as follows:

(5x+3)(x–1)(3x–2)

= [(5x+3)(x–1)](3x–2)

= [5x(x–1)+3(x–1)](3x–2)

= [5x^{2}–5x+3x–3](3x–2)

= 3x(5x^{2}+2x–3)–2(5x^{2}+2x–3)

= 15x^{3}–6x^{2}–9x–[10x^{2}–4x–6]

= 15x^{3}–6x^{2}–9x–10x^{2}+4x+6

= 15x^{3}–16x^{2}–5x+6

Thus, the answer is = 15x^{3}–6x^{2}–9x–10x^{2}+4x+6

= 15x^{3}–16x^{2}–5x+6

**26. (****5****–****x****) (****6****–****5****x****)(****2****–****x****)**

**Solution: **

To simplify, we will use distributive law as follows:

(5–x)(6–5x)(2–x)

=[(5–x)(6–5x)](2–x)

=[5(6–5x)–x(6–5x)](2–x)

=(30–25x–6x+5x^{2})(2–x)

=(30–31x+5x^{2})(2–x)

=2(30–31x+5x^{2})–x(30–31x+5×2)

=60–62x+10x^{2}–30x+31x^{2}–5×3

=60–92x+41x^{2}–5×3

Thus, the answer is 60–92x+41x^{2}–5×3

**27. (****2****x****2****+****3****x****–****5****)(****3****x****2****–****5****x****+****4****)**

**Solution: **

To simplify, we will use distributive law as follows:

(2x^{2}+3x–5)(3×2–5x+4)

=2x^{2}(3x^{2}–5x+4)+3x(3x^{2}–5x+4)–5(3x^{2}–5x+4)

=6x^{4}–10x^{3}+8x^{2}+9x^{3}–15x^{2}+12x–15x^{2}+25x–20

=6x^{4}–10x^{3}+9x^{3}+8x^{2}–15x^{2}–15x^{2}+25x+12x–20

=6x^{4}–x^{3}–22x^{2}+36x–20

Thus, the answer is 6x^{4}–x^{3}–22x^{2}+36x–20

**28. (3x–2) (2x–3)+(5x–3) (x+1)**

**Solution: **

To simplify, we will use distributive law as follows:

(3x–2)(2x–3) + (5x–3)(x+1)

= [(3x–2) (2x–3)] + [(5x–3) (x+1)]

= [3x (2x–3)–2(2x–3)] + [5x(x+1)–3(x+1)]

= 6x^{2}–9x–4x+6+5x^{2}+5x–3x–3

= 6x^{2}+5x^{2}–9x–4x+5x–3x–3+6

= 11x^{2}–11x+3

Thus, the answer is 11x^{2}–11x+3

**29. (****5****x****–****3****) (****x****+****2****)–(****2****x****+****5****)(****4****x****–****3****)**

**Solution: **

To simplify, we will use distributive law as follows:

(5x–3)(x+2)–(2x+5)(4x–3)

= [(5x–3)(x+2)]–[(2x+5)(4x–3)]

= [5x(x+2)–3(x+2)]–[2x(4x–3)+5(4x–3)]

= 5x^{2}+10x–3x–6+8x^{2}+6x–20x+15

= 5x^{2}–8x^{2}+10x–3x+6x−20x–6+15

= −3x^{2}–7x+9

Thus, the answer is −3x^{2}–7x+9.

**30. (****3****x****+****2****y****)(****4****x****+****3****y****)–(****2****x****–****y****)(****7****x****–****3****y****)**

**Solution: **

To simplify, we will use distributive law as follows:

(3x+2y)(4x+3y)–(2x–y)(7x–3y)

=[(3x+2y)(4x+3y)]–[(2x–y)(7x–3y)]

=[3x(4x+3y)+2y(4x+3y)]–[2x(7x–3y)–y(7x–3y)]

=12x^{2}+9xy+8xy+6y^{2}–14x^{2}+6xy+7xy–3y^{2}

=12x^{2}–14x^{2}+9xy+8xy+6xy+7xy+6y2–3y^{2}

=–2x^{2}+30xy+3y^{2}

Thus, the answer is −2x^{2}+30xy+3y^{2}

**31. (****x****2****–****3****x****+****2****) (****5****x****–****2****)–(****3****x****2****+****4****x****–****5****)(****2****x****–****1****)**

**Solution: **

To simplify, we will use distributive law as follows:

(x2–3x+2)(5x–2)–(3×2+4x–5)(2x–1)

= [(x^{2}–3x+2)(5x–2)]–[(3×2+4x–5)(2x–1)]

= [5x(x^{2}–3x+2)–2(x^{2}–3x+2)]–[2x(3x^{2}+4x–5)–1(3x^{2}+4x–5)]

= [5x^{3}–15x^{2}+10x–2x^{2}+6x–4]–[6x^{3}+8x^{2}–10x–3x^{2}–4x+5]

= 5x^{3}–15x^{2}+10x–2x^{2}+6x–4–6x^{3}–8x^{2}+10x+3x^{2}+4x–5

= –x^{3}–22x^{2}+30x–9

Thus, the answer is −x3–22x^{2}+30x–9.

**32. (****x****3****–****2****x****2****+****3****x****–****4****)(****x****–****1****)–(****2****x****–****3****)(****x****2****–****x****+****1****)**

**Solution: **

To simplify, we will use distributive law as follows:

(x^{3}–2x^{2}+3x–4)(x–1)–(2x–3)(x^{2}–x+1)

=[(x^{3}–2x^{2}+3x–4)(x–1)]–[(2x–3)(x^{2}–x+1)]

=[x(x^{3}–2x^{2}+3x–4)–1(x^{3}–2x^{2}+3x–4)]–[2x(x^{2}–x+1)–3(x^{2}–x+1)]

=x4–2x^{3}+3x^{2}–4x–x^{3}+2x^{2}–3x+4–2x^{3}+2x^{2}–2x+3x^{2}–3x+3

=x4–2x^{3}–2×3–x^{3}+3x^{2}+2x^{2}+2x^{2}+3x^{2}–4x–3x–2x–3x+4+3

=x4–5×3+10x^{2}–12x+7

Thus, the answer is x4–5×3+10×2–12x+7..