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)
=(35x2+10x)+(21x+6)
=35x2+10x+21x+6
=35x2+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)
= (2x2–6x) + (8x–24)
= 2x2–6x + 8x–24 = 2x2+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)
=7x2+35xy+xy+5y2
=7x2+36xy+5y2
Thus, the answer is 7x2+36xy+5y2.
4. (a – 1) by (0.1a2 + 3)
Solution:
To multiply, we will use distributive law as follows:
(a–1)(0.1a2+3)
=0.1a2(a–1)+3(a–1)
=0.1a3–0.1a2+3a–3
Thus, the answer is 0.1a3–0.1a2+3a–3.
5. (3x2+y2)(2x2+3y2)
Solution:
To multiply, we will use distributive law as follows:
(3x2+y2)(2x2+3y2)
=3x2(2x2+3y2)+y2(2x2+3y2)
=6x4+9x2y2+2x2y2+3y4
=6x4+11x2y2+3y4
Thus, the answer is 6x4+11x2y2+3y4.
7. (x6–y6) (x2+y2)
Solution
To multiply, we will use distributive law as follows:
(x6–y6) (x2+y2)
=x6(x2+y2) – y6(x2+y2)
=(x8+x6y2) – (y6x2+y8)
=x8+x6y2–y6x2–y8
Thus, the answer is x8+x6y2–y6x2–y8
8. (x2+y2) (3a+2b)
Solution:
To multiply, we will use distributive law as follows:
(x2+y2) (3a+2b)
=x2(3a+2b)+y2(3a+2b)
=3ax2+2bx2+3ay2+2by2
Thus, the answer is 3ax2+2bx2+3ay2+2by2
9. [–3d+(–7f)] (5d+f)
Solution:
To multiply, we will use distributive law as follows:
[–3d+(–7f)](5d+f)
= (–3d)(5d+f) + (–7f)(5d+f)
= (–15d2–3df) + (–35df–7f2)
= –15d2–3df–35df–7f2
= –15d2–38df–7f2
Thus, the answer is −15d2–38df–7f2.
10. (0.8a–0.5b) (1.5a–3b)
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)
= 1.2a2–2.4ab –0.75ab+1.5b2
= 1.2a2–3.15ab+1.5b2
Thus, the answer is 1.2a2–3.15ab+1.5b2.
11. (2x2y2–5xy2)(x2–y2)
Solution:
To multiply, we will use distributive law as follows:
(2x2y2–5xy2) (x2–y2)
= 2x2y2 (x2–y2)–5xy2 (x2–y2)
= 2x4y2–2x2y4–5x3y2+5xy4
Thus, the answer is 2x4y2–2x2y4–5x3y2+5xy4.
14. (3x2y–5xy2)(15x2+13y2)
Solution:
To multiply, we will use distributive law as follows:
(3x2y–5xy2)(15x2+13y2)
=15x2(3x2y–5xy2)+13y2(3x2y–5xy2)
=35x4y–x3y2+x2y3–53xy4
Thus, the answer is 35x4y–x3y2+x2y3–53xy4.
15. (2x2–1) (4x3+5x2)
Solution:
To multiply, we will use distributive law as follows:
(2x2–1) (4x3+5x2)
=2x2(4x3+5x2)–1(4x3+5x2)
=8x5+10x4–4x3–5x2
Thus, the answer is 8x5+10x4–4x3–5x2.
16. (2xy+3y2) (3y2–2)
Solution:
To multiply, we will use distributive law as follows:
(2xy+3y2)(3y2–2)
=2xy(3y2–2)+3y2(3y2–2)
=6xy3–4xy+9y4–6y2
=9y4+6xy3–6y2–4xy
Thus, the answer is 9y4+6xy3–6y2–4xy.
Find the following products and verify the result for x = -1 and y = -2:
17. (3x–5y) (x+y)
Solution:
To multiply, we will use distributive law as follows:
(3x–5y)(x+y)
= 3x(x+y)–5y(x+y)
= 3x2+3xy–5xy–5y2
=3x2–2xy –5y2
= ∴(3x–5y) (x+y)
= 3x2 –2xy –5y2
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 = 3x2 – 2xy–5y2
=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. (x2y–1) (3–2x2y)
Solution:
To multiply, we will use distributive law as follows:
(x2y–1)(3–2x2y)
=x2y(3–2x2y)–1(3–2x2y)
=3x2y–2x4y2–3+22y
=5x2y–2x4y2–3
∴ (x2y–1)(3–2x2y)=5x2y–2x4y2–3
Now, we put x = -1 and y = -2 on both sides to verify the result.
LHS = (x2y–1)(3–2x2y)
= [(−1)2(−2)–1][3–2(−1)2(−2)]
= [1×(−2)–1][3–2×1×(−2)]
= (−2–1)(3+4)
= −3×7
= −21
RHS = 5x2y–2x4y2–3
= 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. x2 (x+2y) (x–3y)
Solution:
To simplify, we will use distributive law as follows:
x2 (x+2y)(x–3y)
=[x2(x+2y)] (x–3y)
=(x3+2x2y)(x–3y)
=x3(x–3y) +2x2y(x–3y)
=x4–3x3+2x3–6x2y2
=x4–x3–6x2y2
Thus, the answer is x4–x3–6x2y2.
21. (x2–2y2)(x+4y)x2y2
Solution:
To simplify, we will use distributive law as follows:
(x2–2y2)(x+4y)x2y2
=[x2(x+4y)–2y2(x+4y)]x2y2
=(x3+4x2y–2xy2–8y3)x2y2
=x5y2+4x4y3–2x3y4–8x2y5
Thus, the answer is x5y2+4x4y3–2x3y4–8x2y5.
22. a2b2 (a+2b)(3a+b)
Solution:
To simplify, we will use distributive law as follows:
a2b2(a+2b) (3a+b)
= [a2b2(a+2b)] (3a+b)
= (a3b2+2a2b3) (3a+b)
= 3a(a3b2+2a2b3)+b(a3b2+2a2b3)
= 3a4b2+6a3b3+a3b3+2a2b4
= 3a4b2+7a3b3+2a2b4
Thus, the answer is 3a4b2+7a3b3+2a2b4.
23. x2 (x–y)y2 (x+2y)
Solution:
To simplify, we will use distributive law as follows:
x2(x–y) y2 (x+2y)
= [x2(x–y)] [y2(x+2y)]
= (x3–x2y) (xy2+2y3)
= x3(xy2+2y3)–x2y(xy2+2y3)
= x4y2+2x3y3–x3y3–2x2y4
= x4y2+x3y3–2x2y4
Thus, the answer is x4y2+x3y3–2x2y4
24. (x3–2x2+5x–7)(2x–3)
Solution:
To simplify, we will use distributive law as follows:
(x3–2x2+5x–7)(2x–3)
=2x(x3–2x2+5x–7)–3(x3–2x2+5x–7)
=2x4–4x3+10x2–14x–3x3+6x2–15x+21
=2x4–4x3–3x3+10x2+6x2–14x–15x+21
=2x4–7x3+16x2–29x+21
Thus, the answer is 2×4–7×3+16×2–29x+21.
25. 2x4–7x3+16x2–29x+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)
= [5x2–5x+3x–3](3x–2)
= 3x(5x2+2x–3)–2(5x2+2x–3)
= 15x3–6x2–9x–[10x2–4x–6]
= 15x3–6x2–9x–10x2+4x+6
= 15x3–16x2–5x+6
Thus, the answer is = 15x3–6x2–9x–10x2+4x+6
= 15x3–16x2–5x+6
26. (5–x) (6–5x)(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+5x2)(2–x)
=(30–31x+5x2)(2–x)
=2(30–31x+5x2)–x(30–31x+5×2)
=60–62x+10x2–30x+31x2–5×3
=60–92x+41x2–5×3
Thus, the answer is 60–92x+41x2–5×3
27. (2x2+3x–5)(3x2–5x+4)
Solution:
To simplify, we will use distributive law as follows:
(2x2+3x–5)(3×2–5x+4)
=2x2(3x2–5x+4)+3x(3x2–5x+4)–5(3x2–5x+4)
=6x4–10x3+8x2+9x3–15x2+12x–15x2+25x–20
=6x4–10x3+9x3+8x2–15x2–15x2+25x+12x–20
=6x4–x3–22x2+36x–20
Thus, the answer is 6x4–x3–22x2+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)]
= 6x2–9x–4x+6+5x2+5x–3x–3
= 6x2+5x2–9x–4x+5x–3x–3+6
= 11x2–11x+3
Thus, the answer is 11x2–11x+3
29. (5x–3) (x+2)–(2x+5)(4x–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)]
= 5x2+10x–3x–6+8x2+6x–20x+15
= 5x2–8x2+10x–3x+6x−20x–6+15
= −3x2–7x+9
Thus, the answer is −3x2–7x+9.
30. (3x+2y)(4x+3y)–(2x–y)(7x–3y)
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)]
=12x2+9xy+8xy+6y2–14x2+6xy+7xy–3y2
=12x2–14x2+9xy+8xy+6xy+7xy+6y2–3y2
=–2x2+30xy+3y2
Thus, the answer is −2x2+30xy+3y2
31. (x2–3x+2) (5x–2)–(3x2+4x–5)(2x–1)
Solution:
To simplify, we will use distributive law as follows:
(x2–3x+2)(5x–2)–(3×2+4x–5)(2x–1)
= [(x2–3x+2)(5x–2)]–[(3×2+4x–5)(2x–1)]
= [5x(x2–3x+2)–2(x2–3x+2)]–[2x(3x2+4x–5)–1(3x2+4x–5)]
= [5x3–15x2+10x–2x2+6x–4]–[6x3+8x2–10x–3x2–4x+5]
= 5x3–15x2+10x–2x2+6x–4–6x3–8x2+10x+3x2+4x–5
= –x3–22x2+30x–9
Thus, the answer is −x3–22x2+30x–9.
32. (x3–2x2+3x–4)(x–1)–(2x–3)(x2–x+1)
Solution:
To simplify, we will use distributive law as follows:
(x3–2x2+3x–4)(x–1)–(2x–3)(x2–x+1)
=[(x3–2x2+3x–4)(x–1)]–[(2x–3)(x2–x+1)]
=[x(x3–2x2+3x–4)–1(x3–2x2+3x–4)]–[2x(x2–x+1)–3(x2–x+1)]
=x4–2x3+3x2–4x–x3+2x2–3x+4–2x3+2x2–2x+3x2–3x+3
=x4–2x3–2×3–x3+3x2+2x2+2x2+3x2–4x–3x–2x–3x+4+3
=x4–5×3+10x2–12x+7
Thus, the answer is x4–5×3+10×2–12x+7..
