15) [tex]$56 x w+49 x k^2-24 y w-21 y k^2$[/tex]
17) [tex]$12 x^2 u+3 x^2 v+28 y u+7 y v$[/tex]
19) [tex]$12 b c-4 b d-15 x c+5 x d$[/tex]
21) [tex]$56 x y-35 x+16 r y-10 r$[/tex]
23) [tex]$5 a^2 z-4 a^2 c+15 x z-12 x c$[/tex]
25) [tex]$21 x y-12 b^2+14 x b-18 b y$[/tex]
16) [tex]$42 m c+36 m d-7 n^2 c-6 n^2 d$[/tex]
18) [tex]$40 a c^2+25 a k^2+32 b c^2+20 b k^2$[/tex]
20) [tex]$16 m n-4 m^2+28 n-7 m$[/tex]
22) [tex]$21 x y+15 x+35 r y+25 r$[/tex]
24) [tex]$4 x y+6-x-24 y$[/tex]
26) [tex]$9 m z-4 n c+3 m c-12 n z$[/tex]
28) [tex]$30 u v+30 u+36 u^2+25 v$[/tex]

Factor each expression by grouping terms with common factors.
Extract the common factors from each group.
Identify and factor out any common binomial factors.
The factorized expressions are: 15) ( 7 x − 3 y ) ( 8 w + 7 k 2 ) , 16) ( 6 m − n 2 ) ( 7 c + 6 d ) , 17) ( 3 x 2 + 7 y ) ( 4 u + v ) , 18) ( 5 a + 4 b ) ( 8 c 2 + 5 k 2 ) , 19) ( 4 b − 5 x ) ( 3 c − d ) , 20) ( 4 m + 7 ) ( 4 n − m ) , 21) ( 7 x + 2 r ) ( 8 y − 5 ) , 22) ( 3 x + 5 r ) ( 7 y + 5 ) , 23) ( a 2 + 3 x ) ( 5 z − 4 c ) , 24) ( x − 6 ) ( 4 y − 1 ) , 25) ( 7 x − 6 b ) ( 3 y + 2 b ) , 26) ( 3 m − 4 n ) ( 3 z + c ) , 28) ( 5 v + 6 u ) ( 6 u + 5 ) .

Explanation

Problem Analysis We are given a list of expressions to factorize. We will use factoring by grouping to factorize each expression.

Factoring 15



56 x w + 49 x k 2 − 24 y w − 21 y k 2 = 7 x ( 8 w + 7 k 2 ) − 3 y ( 8 w + 7 k 2 ) = ( 7 x − 3 y ) ( 8 w + 7 k 2 )


Factoring 16


42 m c + 36 m d − 7 n 2 c − 6 n 2 d = 6 m ( 7 c + 6 d ) − n 2 ( 7 c + 6 d ) = ( 6 m − n 2 ) ( 7 c + 6 d )


Factoring 17


12 x 2 u + 3 x 2 v + 28 y u + 7 y v = 3 x 2 ( 4 u + v ) + 7 y ( 4 u + v ) = ( 3 x 2 + 7 y ) ( 4 u + v )


Factoring 18


40 a c 2 + 25 a k 2 + 32 b c 2 + 20 b k 2 = 5 a ( 8 c 2 + 5 k 2 ) + 4 b ( 8 c 2 + 5 k 2 ) = ( 5 a + 4 b ) ( 8 c 2 + 5 k 2 )


Factoring 19


12 b c − 4 b d − 15 x c + 5 x d = 4 b ( 3 c − d ) − 5 x ( 3 c − d ) = ( 4 b − 5 x ) ( 3 c − d )


Factoring 20


16 mn − 4 m 2 + 28 n − 7 m = 4 m ( 4 n − m ) + 7 ( 4 n − m ) = ( 4 m + 7 ) ( 4 n − m )


Factoring 21


56 x y − 35 x + 16 ry − 10 r = 7 x ( 8 y − 5 ) + 2 r ( 8 y − 5 ) = ( 7 x + 2 r ) ( 8 y − 5 )


Factoring 22


21 x y + 15 x + 35 ry + 25 r = 3 x ( 7 y + 5 ) + 5 r ( 7 y + 5 ) = ( 3 x + 5 r ) ( 7 y + 5 )


Factoring 23


5 a 2 z − 4 a 2 c + 15 x z − 12 x c = a 2 ( 5 z − 4 c ) + 3 x ( 5 z − 4 c ) = ( a 2 + 3 x ) ( 5 z − 4 c )


Factoring 24


4 x y + 6 − x − 24 y = 4 x y − x − 24 y + 6 = x ( 4 y − 1 ) − 6 ( 4 y − 1 ) = ( x − 6 ) ( 4 y − 1 )


Factoring 25


21 x y − 12 b 2 + 14 x b − 18 b y = 21 x y + 14 x b − 18 b y − 12 b 2 = 7 x ( 3 y + 2 b ) − 6 b ( 3 y + 2 b ) = ( 7 x − 6 b ) ( 3 y + 2 b )


Factoring 26


9 m z − 4 n c + 3 m c − 12 n z = 3 m ( 3 z + c ) − 4 n ( c + 3 z ) = ( 3 m − 4 n ) ( 3 z + c )


Factoring 28


30 uv + 30 u + 36 u 2 + 25 v = 6 u ( 6 u + 5 ) + 5 ( 6 u + 5 v ) This expression seems to have a typo. It should be 30 uv + 30 u + 36 u 2 + 25 v . Let's try to factor 30 uv + 36 u 2 + 30 u + 25 v = 6 u ( 5 v + 6 u + 5 ) + 5 ( 5 v ) . It does not lead to a factorization. Let's assume the expression is 30 uv + 25 v + 36 u 2 + 30 u = 5 v ( 6 u + 5 ) + 6 u ( 6 u + 5 ) = ( 5 v + 6 u ) ( 6 u + 5 )


Final Answer The factorized expressions are:


( 7 x − 3 y ) ( 8 w + 7 k 2 )
( 6 m − n 2 ) ( 7 c + 6 d )
( 3 x 2 + 7 y ) ( 4 u + v )
( 5 a + 4 b ) ( 8 c 2 + 5 k 2 )
( 4 b − 5 x ) ( 3 c − d )
( 4 m + 7 ) ( 4 n − m )
( 7 x + 2 r ) ( 8 y − 5 )
( 3 x + 5 r ) ( 7 y + 5 )
( a 2 + 3 x ) ( 5 z − 4 c )
( x − 6 ) ( 4 y − 1 )
( 7 x − 6 b ) ( 3 y + 2 b )
( 3 m − 4 n ) ( 3 z + c )
( 5 v + 6 u ) ( 6 u + 5 )

Examples
Factoring by grouping is a useful technique in many areas of mathematics, including simplifying algebraic expressions, solving equations, and analyzing functions. For example, in physics, you might use factoring to simplify an equation that describes the motion of an object. In computer science, factoring can be used to optimize code by identifying common subexpressions. Understanding factoring helps in problem-solving across various disciplines by allowing you to break down complex expressions into simpler, more manageable parts.

Answered by GinnyAnswer | 2025-07-07

The expressions were factored using the method of grouping, which involves identifying common terms and extracting factors. The final factorized forms for each expression were provided step by step. This technique aids in simplifying algebraic problems and is a foundational skill in mathematics.
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Answered by Anonymous | 2025-07-28