Section 1
Factoring Out the Greatest Common Factor; Factoring by Grouping
Solve each equation and check all solutions.$3 x-2(x+1)=5$
Solve each equation and check all solutions.$5(y-1)+1=y$
Solve each equation and check all solutions.$\frac{2 x-7}{5}=3$
Solve each equation and check all solutions.$2 x-\frac{x}{2}=5 x$
Fill in the blanks.A natural number greater than 1 whose only factors are 1 and itself is called a __________ number.
Fill in the blanks.If a natural number is written as the product of prime numbers, it is written in __________ form.
Fill in the blanks.The GCF of several natural numbers is the __________ number that divides each of the numbers.
Fill in the blanks.The GCF of $6 a^3 b^3-3 a^2 b^2$ is __________.
Fill in the blanks.The GCF of $x(a+b)-y(a+b)$ is __________
Fill in the blanks.Check the results of a factoring problem by __________
Find the prime factorization of each number. 12
Find the prime factorization of each number.24
Find the prime factorization of each number.15
Find the prime factorization of each number.20
Find the prime factorization of each number.40
Find the prime factorization of each number.62
Find the prime factorization of each number.98
Find the prime factorization of each number.112
Find the prime factorization of each number.225
Find the prime factorization of each number.144
Find the prime factorization of each number.288
Find the prime factorization of each number.968
Complete each factorization.$4 a+12=(a+3)$
Complete each factorization.$3 t-27=3(t-)$
Complete each factorization.$r^4+r^2=r^2(+1)$
Complete each factorization.$a^3-a^2=(a-1)$
Complete each factorization.$4 y^2+8 y-2 x y=2 y(2 y+\square)$
Complete each factorization.$3 x^2-6 x y+9 x y^2=\left(-2 y+3 y^2\right)$
Complete each factorization.$a(x+y)+b(x+y)=(x+y)$
Complete each factorization.$p(m-n)-q(m-n)=$
Complete each factorization.$(r-s) p-(r-s) q=(r-s)$
Complete each factorization.$a x+b x+a p+b p=x$ $\square$ $+p$$$=\quad(x+p)$$
Factor out the greatest common factor.$3 x+6$
Factor out the greatest common factor.$2 y-10$
Factor out the greatest common factor.$x y-x z$
Factor out the greatest common factor.$u v+u t$
Factor out the greatest common factor.$t^3+2 t^2$
Factor out the greatest common factor.$b^3-3 b^2$
Factor out the greatest common factor.$r^4-r^2$
Factor out the greatest common factor.$a^3+a^2$
Factor out the greatest common factor.$a^3 b^3 z^3-a^2 b^3 z^2$
Factor out the greatest common factor.$r^3 s^6 t^9+r^2 s^2 t^2$
Factor out the greatest common factor.$24 x^2 y^3 z^4+8 x y^2 z^3$
Factor out the greatest common factor.$3 x^2 y^3-9 x^4 y^3 z$
Factor out the greatest common factor.$12 u v w^3-18 u v^2 w^2$
Factor out the greatest common factor.$14 x y z-16 x^2 y^2 z$
Factor out the greatest common factor.$3 x+3 y-6 z$
Factor out the greatest common factor.$2 x-4 y+8 z$
Factor out the greatest common factor.$a b+a c-a d$
Factor out the greatest common factor.$r s-r t+r u$
Factor out the greatest common factor.$4 y^2+8 y-2 x y$
Factor out the greatest common factor.$3 x^2-6 x y+9 x y^2$
Factor out the greatest common factor.$12 r^2-3 r s+9 r^2 s^2$
Factor out the greatest common factor.$6 a^2-12 a^3 b+36 a b$
Factor out the greatest common factor.$a b x-a b^2 x+a b x^2$
Factor out the greatest common factor.$a^2 b^2 x^2+a^3 b^2 x^2-a^3 b^3 x^3$
Factor out the greatest common factor.$4 x^2 y^2 z^2-6 x y^2 z^2+12 x y z^2$
Factor out the greatest common factor.$32 x y z+48 x^2 y z+36 x y^2 z$
Factor out the greatest common factor.$70 a^3 b^2 c^2+49 a^2 b^3 c^3-21 a^2 b^2 c^2$
Factor out the greatest common factor.$8 a^2 b^2-24 a b^2 c+9 b^2 c^2$
Factor out -1 from each polynomial.$-a-b$
Factor out -1 from each polynomial.$-x-2 y$
Factor out -1 from each polynomial.$-2 x+5 y$
Factor out -1 from each polynomial.$-3 x+8 z$
Factor out -1 from each polynomial.$-2 a+3 b$
Factor out -1 from each polynomial.$-3 m-4 n+1$
Factor out -1 from each polynomial.$-3 r+2 s-3$
Factor out -1 from each polynomial.$-3 x y+2 z+5 w$
Factor out -1 from each polynomial.$-4 a b+3 c-5 d$
Factor out -1 from each polynomial.$-3 a b-5 a c+9 b c$
Factor out -1 from each polynomial.$-6 y z+12 x z-5 x y$
Factor out the greatest common factor, including -1.$-3 x^2 y-6 x y^2$
Factor out the greatest common factor, including -1.$-4 a^2 b^2+6 a b^2$
Factor out the greatest common factor, including -1.$-4 a^2 b^3+12 a^3 b^2$
Factor out the greatest common factor, including -1.$-25 x^4 y^3 z^2+30 x^2 y^3 z^4$
Factor out the greatest common factor, including -1.$-4 a^2 b^2 c^2+14 a^2 b^2 c-10 a b^2 c^2$
Factor out the greatest common factor, including -1.$-10 x^4 y^3 z^2+8 x^3 y^2 z-20 x^2 y$
Factor out the greatest common factor, including -1.$-14 a^6 b^6+49 a^2 b^3-21 a b$
Factor out the greatest common factor, including -1.$-35 r^9 s^9 t^9+25 r^6 s^6 t^6+75 r^3 s^3 t^3$
Factor out the greatest common factor, including -1.$-5 a^2 b^3 c+15 a^3 b^4 c^2-25 a^4 b^3 c$
Factor out the greatest common factor, including -1.$-7 x^5 y^4 z^3+49 x^5 y^5 z^4-21 x^6 y^4 z^3$
Factor each expression.$(x+y) 2+(x+y) b$
Factor each expression.$(a-b) c+(a-b) d$
Factor each expression.$3(x+y)-a(x+y)$
Factor each expression.$x(y+1)-5(y+1)$
Factor each expression.$3(r-2 s)-x(r-2 s)$
Factor each expression.$x(a+2 b)+y(a+2 b)$
Factor each expression.$(x-3)^2+(x-3)$
Factor each expression.$(3 t+5)^2-(3 t+5)$
Factor each expression.$2 x\left(a^2+b\right)+2 y\left(a^2+b\right)$
Factor each expression.$3 x(c-3 d)+6 y(c-3 d)$
Factor each expression.$3 x^2(r+3 s)-6 y^2(r+3 s)$
Factor each expression.$9 a^2 b^2(3 x-2 y)-6 a b(3 x-2 y)$
Factor each expression.$3 x(a+b+c)-2 y(a+b+c)$
Factor each expression.$2 m(a-2 b+3 c)-21 x y(a-2 b+3 c)$
Factor each expression.$14 x^2 y(r+2 s-t)-21 x y(r+2 s-t)$
Factor each expression.$15 x y^3(2 x-y+3 z)+25 x y^2(2 x-y+3 z)$
Factor each expression.$(x+3)(x+1)-y(x+1)$
Factor each expression.$x\left(x^2+2\right)-y\left(x^2+2\right)$
Factor each expression.$(3 x-y)\left(x^2-2\right)+\left(x^2-2\right)$
Factor each expression.$(x-5 y)(a+2)-(x-5 y)$
Factor each expression.$2 x+2 y+a x+a y$
Factor each expression.$b x+b z+5 x+5 z$
Factor each expression.$7 r+7 s-k r-k s$
Factor each expression.$9 p-9 q+m p-m q$
Factor each expression.$x r+x s+y r+y s$
Factor each expression.$p m-p n+q m-q n$
Factor each expression.$2 a x+2 b x+3 a+3 b$
Factor each expression.$3 x y+3 x z-5 y-5 z$
Factor each expression.$2 a b+2 a c+3 b+3 c$
Factor each expression.$3 a c+a+3 b c+b$
Factor each expression.$2 x^2+2 x y-3 x-3 y$
Factor each expression.$3 a b+9 a-2 b-6$
Factor each expression.$3 t v-9 t w+u v-3 u w$
Factor each expression.$c e-2 c f+3 d e-6 d f$
Factor each expression.$9 m p+3 m q-3 n p-n q$
Factor each expression.$a x+b x-a-b$
Factor each expression.$m p-n p-m+n$
Factor each expression.$6 x^2 u-3 x^2 v+2 y u-y v$
Factor each expression.$x(a-b)+y(b-a)$
Factor each expression.$p(m-n)-q(n-m)$
Factor each expression. Factor out all common factors first, if they exist.$a x^3+b x^3+2 a x^2 y+2 b x^2 y$
Factor each expression. Factor out all common factors first, if they exist.$x^3 y^2-2 x^2 y^2+3 x y^2-6 y^2$
Factor each expression. Factor out all common factors first, if they exist.$4 a^2 b+12 a^2-8 a b-24 a$
Factor each expression. Factor out all common factors first, if they exist.$-4 a b c-4 a c^2+2 b c+2 c^2$
Factor each expression. Factor out all common factors first, if they exist.$x^3+2 x^2+x+2$
Factor each expression. Factor out all common factors first, if they exist.$y^3-3 y^2-5 y+15$
Factor each expression. Factor out all common factors first, if they exist.$x^3 y-x^2 y-x y^2+y^2$
Factor each expression. Factor out all common factors first, if they exist.$2 x^3 z-4 x^2 z+32 x z-64 z$
Factor each expression completely. You may have to rearrange some terms first.$2 r-b s-2 s+b r$
Factor each expression completely. You may have to rearrange some terms first.$5 x+r y+r x+5 y$
Factor each expression completely. You may have to rearrange some terms first.$a x+b y+b x+a y$
Factor each expression completely. You may have to rearrange some terms first.$m r+n s+m s+n r$
Factor each expression completely. You may have to rearrange some terms first.$a c+b d-a d-b c$
Factor each expression completely. You may have to rearrange some terms first.$s x-r y+r x-s y$
Factor each expression completely. You may have to rearrange some terms first.$a r^2-b r s+a r s-b r^2$
Factor each expression completely. You may have to rearrange some terms first.$a^2 b c+a^2 c+a b c+a c$
Factor each expression completely. You may have to rearrange some terms first.$b a+3+a+3 b$
Factor each expression completely. You may have to rearrange some terms first.$x y+7+y+7 x$
Factor each expression completely. You may have to rearrange some terms first.$p r+q s-p s-q r$
Factor each expression completely. You may have to rearrange some terms first.$a c-b d-a d+b c$
When we add $5 x$ and $7 x$, we combine like terms: $5 x+7 x=12 x$. Explain how this is related to factoring out a common factor.
Explain how you would factor $x(a-b)+y(b-a)$.
Think of two positive integers. Divide their product by their greatest common factor. Why do you think the result is called the lowest common multiple of the two integers?
Two integers are relatively prime if their greatest common factor is 1 . For example, 6 and 25 are relatively prime, but 6 and 15 are not. If the greatest common factor of three integers is 1 , must any two of them be relatively prime? Explain.
Factor $a x+a y+b x+b y$ by grouping the first two terms and the last two terms. Then rearrange the terms as $a x+b x+a y+b y$, and factor again by grouping the first two and the last two. Do the results agree?
Factor $2 x y+2 x z-3 y-3 z$ by grouping in two different ways.