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Section 1
Greatest Common Factor and Factoring by Grouping
The final exam is just around the corner. Your old tests and quizzes provide good material to study for the final exam. Use your old tests to make a list of the chapters on which you need to concentrate. Ask your professor for help if there are still concepts that you do not understand.
Define the key terms.a. factoringb. greatest common factor (GCF)c. prime factorizationd. prime polynomial
Identify the greatest common factor.$$28,63$$
Identify the greatest common factor.$$24,40$$
Identify the greatest common factor.$$42,30,60$$
Identify the greatest common factor.$$20,52,32$$
Identify the greatest common factor.$$3 x y, 7 y$$
Identify the greatest common factor.$$10 \mathrm{mm}, 11 n$$
Identify the greatest common factor.$$12 w^{3} z, 16 w^{2} z$$
Identify the greatest common factor.$$20 c d, 15 c^{3} d$$
Identify the greatest common factor.$$8 x^{3} y^{4} z^{2}, 12 x y^{5} z^{4}, 6 x^{2} y^{x} z^{3}$$
Identify the greatest common factor.$$15 r^{2} s^{2} t^{5}, 5 r^{3} s^{4} t^{3}, 30 r^{4} s^{3} t^{2}$$
Identify the greatest common factor.$$7(x-y), 9(x-y)$$
Identify the greatest common factor.$$(2 a-b), 3(2 a-b)$$
a. Use the distributive property to multiply $3(x-2 y)$b. Use the distributive property to factor $3 x-6 y$
a. Use the distributive property to multiply $a^{2}(5 a+b)$b. Use the distributive property to factor $5 a^{3}+a^{2} b$
Factor out the GCF.$$4 p+12$$
Factor out the GCF.$$3 q-15$$
Factor out the GCF.$$5 c^{2}-10 c+15$$
Factor out the GCF.$$16 d^{3}+24 d^{2}+32 d$$
Factor out the GCF.$$x^{5}+x^{3}$$
Factor out the GCF.$$y^{2}-y^{3}$$
Factor out the GCF.$$t^{4}-4 t+8 t^{2}$$
Factor out the GCF.$$7 r^{3}-r^{5}+r^{4}$$
Factor out the GCF.$$2 a b+4 a^{3} b$$
Factor out the GCF.$$5 u^{3} v^{2}-5 u v$$
Factor out the GCF.$$38 x^{2} y-19 x^{2} y^{4}$$
Factor out the GCF.$$100 a^{5} b^{3}+16 a^{2} b$$
Factor out the GCF.$$6 x^{3} y^{5}-18 x y^{9} z$$
Factor out the GCF.$$15 m p^{7} q^{4}+12 m^{4} q^{3}$$
Factor out the GCF.$$5+7 y^{3}$$
Factor out the GCF.$$w^{3}-5 u^{3} v^{2}$$
Factor out the GCF.$$42 p^{3} q^{2}+14 p q^{2}-7 p^{4} q^{4}$$
Factor out the GCF.$$8 m^{2} n^{3}-24 m^{2} n^{2}+4 m^{3} n$$
Factor out the GCF.$$t^{5}+2 r t^{3}-3 t^{4}+4 r^{2} t^{2}$$
Factor out the GCF.$$u^{2} v+5 u^{3} v^{2}-2 u^{2}+8 u v$$
For the polynomial $-2 x^{3}-4 x^{2}+8 x$a. Factor out $-2 x$b. Factor out $2 x$
For the polynomial $-9 y^{5}+3 y^{3}-12 y$a. Factor out $-3 y$b. Factor out $3 y$
Factor out $-1$ from the polynomial $-8 t^{2}-9 t-2$
Factor out $-1$ from the polynomial $-6 x^{3}-2 x-5$
Factor out the opposite of the greatest common factor.$$-15 p^{3}-30 p^{2}$$
Factor out the opposite of the greatest common factor.$$-24 m^{3}-12 m^{4}$$
Factor out the opposite of the greatest common factor.$$-3 m^{4} n^{2}+6 m^{2} n-9 m n^{2}$$
Factor out the opposite of the greatest common factor.$$-12 p^{3} t+2 p^{2} t^{3}+6 p t^{2}$$
Factor out the opposite of the greatest common factor.$$-7 x-6 y-2 z$$
Factor out the opposite of the greatest common factor.$$-4 a+5 b-c$$
Factor out the GCF.$$13(a+6)-4 b(a+6)$$
Factor out the GCF.$$7\left(x^{2}+1\right)-y\left(x^{2}+1\right)$$
Factor out the GCF.$$8 v\left(w^{2}-2\right)+\left(w^{2}-2\right)$$
Factor out the GCF.$$t(r+2)+(r+2)$$
Factor out the GCF.$$21 x(x+3)+7 x^{2}(x+3)$$
Factor out the GCF.$$5 y^{3}(y-2)-15 y(y-2)$$
Factor by grouping.$$8 a^{2}-4 a b+6 a c-3 b c$$
Factor by grouping.$$4 x^{3}+3 x^{2} y+4 x y^{2}+3 y^{3}$$
Factor by grouping.$$3 q+3 p+q r+p r$$
Factor by grouping.$$x y-x z+7 y-7 z$$
Factor by grouping.$$6 x^{2}+3 x+4 x+2$$
Factor by grouping.$$4 y^{2}+8 y+7 y+14$$
Factor by grouping.$$2 t^{2}+6 t-t-3$$
Factor by grouping.$$2 p^{2}-p-2 p+1$$
Factor by grouping.$$6 y^{2}-2 y-9 y+3$$
Factor by grouping.$$5 a^{2}+30 a-2 a-12$$
Factor by grouping.$$b^{4}+b^{3}-4 b-4$$
Factor by grouping.$$8 w^{5}+12 w^{2}-10 w^{3}-15$$
Factor by grouping.$$3 j^{2} k+15 k+j^{2}+5$$
Factor by grouping.$$2 a b^{2}-6 a c+b^{2}-3 c$$
Factor by grouping.$$14 w^{6} x^{6}+7 w^{6}-2 x^{6}-1$$
Factor by grouping.$$18 p^{4} q-9 p^{5}-2 q+p$$
Factor by grouping.$$a y+b x+b y+a x$$(Hint: Rearrange the terms.)
Factor by grouping.$$2 c+3 a y+a c+6 y$$
Factor by grouping.$$v w^{2}-3+w-3 w v$$
Factor by grouping.$$2 x^{2}+6 m+12+x^{2} m$$
Factor out the GCF first. Then factor by grouping.$$15 x^{4}+15 x^{2} y^{2}+10 x^{3} y+10 x y^{3}$$
Factor out the GCF first. Then factor by grouping.$$2 a^{3} b-4 a^{2} b+32 a b-64 b$$
Factor out the GCF first. Then factor by grouping.$$4 a b x-4 b^{2} x-4 a b+4 b^{2}$$
Factor out the GCF first. Then factor by grouping.$$p^{2} q-p q^{2}-r p^{2} q+r p q^{2}$$
Factor out the GCF first. Then factor by grouping.$$6 s t^{2}-18 s t-6 t^{4}+18 t^{3}$$
Factor out the GCF first. Then factor by grouping.$$15 j^{3}-10 j^{2} k-15 j^{2} k^{2}+10 j k^{3}$$
The formula $P=2 l+2 w$ represents the perimeter, $P,$ of a rectangle given the length, $l$ and the width, $w$. Factor out the GCF and write an equivalent formula in factored form.
The formula $P=2 a+2 b$ represents the perimeter, $P,$ of a parallelogram given the base, $b$ and an adjacent side, $a$. Factor out the GCF and write an equivalent formula in factored form.
The formula $S=2 \pi r^{2}+2 \pi r h$ represents the surface area, $S,$ of a cylinder with radius, $r,$ and height, $h$. Factor out the GCF and write an equivalent formula in factored form.
The formula $A=P+P r t$ represents the total amount of money, $A,$ in an account that earns simple interest at a rate, $r,$ for $t$ years. Factor out the GCF and write an equivalent formula in factored form.
Factor out $\frac{1}{7}$ from $\frac{1}{7} x^{2}+\frac{3}{7} x-\frac{5}{7}$
Factor out $\frac{1}{5}$ from $\frac{6}{5} y^{2}-\frac{4}{5} y+\frac{1}{5}$
Factor out $\frac{1}{4}$ from $\frac{5}{4} w^{2}+\frac{3}{4} w+\frac{9}{4}$
Factor out $\frac{1}{6}$ from $\frac{1}{6} p^{2}-\frac{3}{6} p+\frac{5}{6}$
Write a polynomial that has a GCF of $3 x$ (Answers may vary.)
Write a polynomial that has a GCF of $7 y .$ (Answers may vary.)
Write a polynomial that has a GCF of $4 p^{2} q$. (Answers may vary.)
Write a polynomial that has a GCF of $2 a b^{2}$. (Answers may vary.)