Deck 8: Factors and Factoring

Question 1

Factor $2 a b^{2}+3 a^{2} b$ and check your results.

Answer 1

$a b(2 b+3 a)$

Question 2

Factor $14 x-35 y+7 z$ and check your results.

Answer 2

$7(2 x-5 y+z)$

Question 3

Factor $15 x^{3}+27 a x^{2}+24 x$ and check your results.

Answer 3

$3 x\left(5 x^{2}+9 a x+8\right)$

Question 4

Factor $2 x^{2}(a-5)-6 x(a-5)^{2}$ and check your results.

Question 5

Factor $9 x y z+12 x^{2} y+6 a x^{3} y$ and check your results.

Question 6

Factor $6 a^{4} b-8 a^{3} b+2 a^{2} b$ and check your results.

Question 7

Factor $\frac{x^{3} y}{a^{2} b}-\frac{x^{2} y^{2}}{a^{2} b^{2}}+\frac{x y z}{a b}$ and check your results.

Question 8

Factor $\frac{3}{x b}+\frac{1}{x b^{2}}+\frac{w}{x^{2} b}$ and check your results.

Question 9

Factor $2 x^{2}+4 x^{4}+16 x^{5}$ and check your results.

Question 10

Factor abcxy $-a^{2} b^{2} c^{2} x^{2} y^{2}+a^{3} b^{3} c^{3} x^{3} y$ and check your results.

Question 11

Factor $6 a^{3} b^{4} c^{4} d^{2}-8 a^{4} b^{4} c^{8} d+2 a^{4} b^{3} c^{7} d^{5}$ and check your results.

Question 12

Factor $4 x^{3} y^{4} z^{2}-8 x^{4} y^{8} z+2 x y$ and check your results.

Question 13

The price of an item costing $P$ dollars is raised by $22 \%$ . The resulting price is $P+0.22 P$ . Factor this expression.

Question 14

A point charge creates an electrical potential. The difference in potential $V$ between a point a distance of $r_{1}$ from the charge and a point a distance of $r_{2}$ from the charge is given by $V=\frac{k q}{r_{2}}-\frac{k q}{r_{1}}$ . Factor the right-hand side.

Question 15

When the radius of a cylinder is increased by 2.0, the surface area becomes $2 \pi(r+2.0)^{2}+2 \pi(r+2.0) h$ . Factor this expression.

Question 16

A cantilever beam is $y$ metres long and has a concentrated load $P$ at the free end. The deflection at the point $x$ metres from the fixed end is given by $\frac{P y x^{2}}{2 E I}-\frac{P x^{3}}{6 E I}$ , where $E$ and $I$ are constants. Factor this expression.

Question 17

A simply-supported beam is $y$ metres long and has a concentrated load $P$ in the centre. The deflection at the point $x$ metres from one end is given by $\frac{P x y^{2}}{16 E I}-\frac{P x^{3}}{12 E I}$ , where $E$ and $I$ are constants. Factor this expression.

Question 18

Factor completely: $16-x^{2}$

Question 19

Factor completely: $x^{2}-49$

Question 20

Factor completely: $16 c^{2}-9 x^{2}$

Question 21

Factor completely: $x^{2} z^{2}-x^{2} y^{2}$

Question 22

Factor completely: $16 c^{2} d^{2}-25 x^{2}$

Question 23

Factor completely: $225 x^{2} y^{2}-1$

Question 24

Factor completely: $a^{3}-64 a x^{2}$

Question 25

Factor completely: $2 x^{3}-8 x$

Question 26

Factor completely: $x^{6}-y^{2}$

Question 27

Factor completely: $16 a^{2} b^{4}-1$

Question 28

Factor completely: $121 x^{2} y^{2}-x^{4}$

Question 29

Factor completely: $36 z^{10}-49 y^{8}$

Question 30

Factor completely: $75-48 z^{10}$

Question 31

Factor completely: $12 a^{6}-48 a^{2} b^{2}$

Question 32

Factor completely: $\left(a^{2}-b^{2}\right)^{2}-x^{2}$

Question 33

Factor completely: $(x-8 y)^{2}-1$

Question 34

Factor completely: $12 r^{2}-3 y^{2}$

Question 35

Factor completely: $(3 x+1)^{2}-4 x^{2}$

Question 36

Factor completely: $4 x^{2}-4 y^{2}$

Question 37

Factor completely: $(4 x)^{2}-4 y^{2}$

Question 38

Factor completely: $9 y^{2}-(4 y+5)^{2}$

Question 39

The vertical sides of a square are increased by $k$ and the horizontal sides are decreased by $k$ . The resulting area is 9 less than the original area. What is the number $k$ ?

Question 40

A simply-supported beam is $y$ metres long and has a couple moment $M$ at one end. The deflection at the point $x$ metres from the other end is given by $\frac{M x y}{6 E I}\left(1-\frac{x^{2}}{y^{2}}\right)$ , where $E$ and $I$ are constants. Factor this expression.

Question 41

Test $3 y^{2}-3 y-5$ for factorability.

Question 42

Test $25 x^{2}-10 x+1$ for factorability.

Question 43

Test $x^{2}-2 x-1$ for factorability.

Question 44

Test $4 x^{2}-5 x+1$ for factorability.

Question 45

Test $-2 x^{2}-3 x+10$ for factorability.

Question 46

Factor completely: $x^{2}-5 x-14$

Question 47

Factor completely: $a^{2}-2 a-3$

Question 48

Factor completely: $c^{2}+13 c+36$

Question 49

Factor completely: $z^{2}+z-132$

Question 50

Factor completely: $x^{2}-8 x-33$

Question 51

Factor completely: $x^{2}+16 x+55$

Question 52

Factor completely: $x^{2}+4 x-12$

Question 53

Factor completely: $x^{2}-23 x+60$

Question 54

Factor completely: $z^{2}-11 c z x+18 c^{2} x^{2}$

Question 55

Factor completely: $c^{2} w^{2}-35 c w y-36 y^{2}$

Question 56

Factor completely: $c^{4 k}+12 c^{2 k}+20$

Question 57

Factor completely: $(b-2)^{2}-2 a(b-2)-3 a^{2}$

Question 58

Factor completely: $(a-b)^{10}+2(a-b)^{5} c-15 c^{2}$

Question 59

To make a box with a square base start with a square piece of cardboard, cut a square of side length $x$ from each corner, and fold up the sides. If the area of the base of the box is $4 x^{2}-40 x+100$ , factor to determine the original side length.

Question 60

To make a box with a rectangular base start with a rectangular piece of cardboard, cut a square of side length $x$ from each corner, and fold up the sides. If the volume of the box is $4 x^{3}-48 x^{2}+140 x$ , factor to determine the original dimensions of the rectangle.

Question 61

Factor $\frac{7 a b}{x}+\frac{a^{2} b}{x^{2}}-\frac{8 a b^{2}}{x^{3}}$ and check your results.

Question 62

Factor completely: $y^{3}-2 y^{2}-a y+2 a$

Question 63

Factor completely: $w x+w y-2 x-2 y$

Question 64

Factor completely: $x y+5 y+3 x+15$

Question 65

Factor completely: $7 x+7 c-a x-a c$

Question 66

Factor completely: $2-4 y-6 x+12 x y$

Question 67

Factor completely: $12 x-15+8 a x-10 a$

Question 68

Factor completely: $6 x y-4 x b-9 a y+6 a b$

Question 69

Factor completely: $x^{2} y^{2}+5 x^{2}+3 y^{2}+15$

Question 70

Factor completely: $x^{2}-3 x z+7 x y-21 y z$

Question 71

Factor completely: $2 x^{5}-x^{4}-2 x y^{4}+y^{4}$

Question 72

Factor completely: $x^{2}+2 x-9 a^{2}+1$

Question 73

Factor completely: $2 a^{3} x+8 a b x^{2}-3 a^{2} b^{2}-12 b^{3} x$

Question 74

Factor completely: $a^{3} x^{3}-4 a^{2} b x y^{3}+5 a b^{3} x^{2} y-20 b^{4} y^{4}$

Question 75

Factor completely: $x y z+x^{2} y z-3 w^{2}-3 x w^{2}$

Question 76

Factor completely: $2 x^{4} z+2 x y^{3}-3 x^{3} y z-3 y^{4}$

Question 77

Factor completely: $x^{2}-2 a x+a^{2}-b^{2}$

Question 78

Factor completely: $3 x^{2}-2 x-8$

Question 79

Factor completely: $2 y^{2}+3 y-2$

Question 80

Factor completely: $2 x^{2}+13 x+15$