Exercise 5.3: Factors of Algebraic Expressions
Find the factors:
1. $x^2 + xy + zx + yz$
Solution: $x^2 + xy + zx + yz$
$= x(x + y) + z(x + y)$
$= (x + y)(x + z)$
Answer:$(x + y)(x + z)$
2. $a^2 + bc + ca + ab$
Solution: $a^2 + bc + ca + ab$
$= a^2 + ab + ca + bc$
$= a(a + b) + c(a + b)$
$= (a + b)(a + c)$
Answer:$(a + b)(a + c)$
3. $ab(px + qy) + a^2qx + b^2py$
Solution: $ab(px + qy) + a^2qx + b^2py$
$= abpx + abqy + a^2qx + b^2py$
$= abpx + a^2qx + abqy + b^2py$
$= ax(bp + aq) + by(aq + bp)$
$= ax(bp + aq) + by(bp + aq)$
$= (bp + aq)(ax + by)$
Answer:$(bp + aq)(ax + by)$
4. $4x^2 – y^2$
Solution: $4x^2 – y^2$
$= (2x)^2 – (y)^2$
$= (2x + y)(2x – y)$
Answer:$(2x + y)(2x – y)$
5. $9a^2 – 4b^2$
Solution: $9a^2 – 4b^2$
$= (3a)^2 – (2b)^2$
$= (3a + 2b)(3a – 2b)$
Answer:$(3a + 2b)(3a – 2b)$
6. $a^2b^2 – 49y^2$
Solution: $a^2b^2 – 49y^2$
$= (ab)^2 – (7y)^2$
$= (ab + 7y)(ab – 7y)$
Answer:$(ab + 7y)(ab – 7y)$
7. $16x^4 – 81y^4$
Solution: $16x^4 – 81y^4$
$= (4x^2)^2 – (9y^2)^2$
$= (4x^2 + 9y^2)(4x^2 – 9y^2)$
$= (4x^2 + 9y^2)\{(2x)^2 – (3y)^2\}$
$= (4x^2 + 9y^2)(2x + 3y)(2x – 3y)$
Answer:$(4x^2 + 9y^2)(2x + 3y)(2x – 3y)$
8. $a^2 – (x + y)^2$
Solution: $a^2 – (x + y)^2$
$= \{a + (x + y)\}\{a – (x + y)\}$
$= (a + x + y)(a – x – y)$
Answer:$(a + x + y)(a – x – y)$
9. $(2x – 3y + 5z)^2 – (x – 2y + 3z)^2$
Solution: $(2x – 3y + 5z)^2 – (x – 2y + 3z)^2$
$= \{(2x – 3y + 5z) + (x – 2y + 3z)\}\{(2x – 3y + 5z) – (x – 2y + 3z)\}$
$= (2x – 3y + 5z + x – 2y + 3z)(2x – 3y + 5z – x + 2y – 3z)$
$= (3x – 5y + 8z)(x – y + 2z)$
Answer:$(3x – 5y + 8z)(x – y + 2z)$
10. $4 + 8a^2 + 9a^4$
Solution: $4 + 8a^2 + 9a^4$
$= 9a^4 + 8a^2 + 4$
$= (3a^2)^2 + 2 \cdot 3a^2 \cdot 2 + 2^2 – 4a^2$
$= (3a^2 + 2)^2 – (2a)^2$
$= (3a^2 + 2 + 2a)(3a^2 + 2 – 2a)$
$= (3a^2 + 2a + 2)(3a^2 – 2a + 2)$
Answer:$(3a^2 + 2a + 2)(3a^2 – 2a + 2)$
11. $2a^2 + 6a – 80$
Solution: $2a^2 + 6a – 80$
$= 2(a^2 + 3a – 40)$
$= 2(a^2 + 8a – 5a – 40)$
$= 2\{a(a + 8) – 5(a + 8)\}$
$= 2(a + 8)(a – 5)$
Answer:$2(a + 8)(a – 5)$
12. $y^2 – 6y – 91$
Solution: $y^2 – 6y – 91$
$= y^2 – 13y + 7y – 91$
$= y(y – 13) + 7(y – 13)$
$= (y – 13)(y + 7)$
Answer:$(y – 13)(y + 7)$
13. $p^2 – 15p + 56$
Solution: $p^2 – 15p + 56$
$= p^2 – 8p – 7p + 56$
$= p(p – 8) – 7(p – 8)$
$= (p – 8)(p – 7)$
Answer:$(p – 8)(p – 7)$
14. $45a^8 – 5a^4x^4$
Solution: $45a^8 – 5a^4x^4$
$= 5a^4(9a^4 – x^4)$
$= 5a^4\{(3a^2)^2 – (x^2)^2\}$
$= 5a^4(3a^2 + x^2)(3a^2 – x^2)$
Answer:$5a^4(3a^2 + x^2)(3a^2 – x^2)$
15. $a^2 + 3a – 40$
Solution: $a^2 + 3a – 40$
$= a^2 + 8a – 5a – 40$
$= a(a + 8) – 5(a + 8)$
$= (a + 8)(a – 5)$
Answer:$(a + 8)(a – 5)$
16. $(x^2 + 1)^2 – (y^2 + 1)^2$
Solution: $(x^2 + 1)^2 – (y^2 + 1)^2$
$= \{(x^2 + 1) + (y^2 + 1)\}\{(x^2 + 1) – (y^2 + 1)\}$
$= (x^2 + 1 + y^2 + 1)(x^2 + 1 – y^2 – 1)$
$= (x^2 + y^2 + 2)(x^2 – y^2)$
$= (x^2 + y^2 + 2)(x + y)(x – y)$
Answer:$(x^2 + y^2 + 2)(x + y)(x – y)$
17. $x^2 + 11x + 30$
Solution: $x^2 + 11x + 30$
$= x^2 + 6x + 5x + 30$
$= x(x + 6) + 5(x + 6)$
$= (x + 6)(x + 5)$
Answer:$(x + 6)(x + 5)$
18. $a^2 – b^2 + 2bc – c^2$
Solution: $a^2 – b^2 + 2bc – c^2$
$= a^2 – (b^2 – 2bc + c^2)$
$= a^2 – (b – c)^2$
$= \{a + (b – c)\}\{a – (b – c)\}$
$= (a + b – c)(a – b + c)$
Answer:$(a + b – c)(a – b + c)$
19. $144x^7 – 25x^3a^4$
Solution: $144x^7 – 25x^3a^4$
$= x^3(144x^4 – 25a^4)$
$= x^3\{(12x^2)^2 – (5a^2)^2\}$
$= x^3(12x^2 + 5a^2)(12x^2 – 5a^2)$
Answer:$x^3(12x^2 + 5a^2)(12x^2 – 5a^2)$
20. $4x^2 + 12xy + 9y^2 – 16a^2$
Solution: $4x^2 + 12xy + 9y^2 – 16a^2$
$= \{(2x)^2 + 2 \cdot 2x \cdot 3y + (3y)^2\} – (4a)^2$
$= (2x + 3y)^2 – (4a)^2$
$= (2x + 3y + 4a)(2x + 3y – 4a)$
Answer:$(2x + 3y + 4a)(2x + 3y – 4a)$