a) [tex]$4 x^4+81$[/tex]
b) [tex]$x^4+625 y^4$[/tex]
c) [tex]$a^4+a^2 b^2+b^4$[/tex]
d) [tex]$a^4+2 a^2+9$[/tex]
e) [tex]$x^4-39 x^2+25$[/tex]
f) [tex]$x^4 y^4+x^2 y^2+1$[/tex]
g) [tex]$8 x^3+27$[/tex]
h) [tex]$135 a^6 x+5 a^3 x^4$[/tex]
i) [tex]$16 a^5 b-54 a^2 b^4$[/tex]
j) [tex]$a^2-8 a-9+10 b-b^2$[/tex]
k) [tex]$x^2-10 x+24+6 y-9 y^2$[/tex]
l) [tex]$\left(1-x^2\right)\left(1-y^2\right)+4 x y$[/tex]
m) [tex]$\left(x^2-4\right)\left(y^2-9\right)-24 x y$[/tex]
Answer (2)
The provided expressions can be factored using various algebraic techniques such as recognizing sums of squares, cubes, and completing the square. Each expression is approached systematically to reveal its factors, from basic algebraic identities to more complex forms. Understanding these strategies is crucial for effective factorization in mathematical contexts.
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Factor each expression by identifying patterns such as difference of squares, sum/difference of cubes, or by completing the square.
Apply algebraic manipulations to rewrite each expression into a factorable form.
Factor each expression completely into its irreducible factors.
The factorized forms are: a) ( 2 x 2 + 6 x + 9 ) ( 2 x 2 − 6 x + 9 ) , b) ( x 2 + 5 2 x y + 25 y 2 ) ( x 2 − 5 2 x y + 25 y 2 ) , c) ( a 2 + ab + b 2 ) ( a 2 − ab + b 2 ) , d) ( a 2 + 2 a + 3 ) ( a 2 − 2 a + 3 ) , e) ( x 2 − 29 x − 5 ) ( x 2 + 29 x − 5 ) , f) ( x 2 y 2 + x y + 1 ) ( x 2 y 2 − x y + 1 ) , g) ( 2 x + 3 ) ( 4 x 2 − 6 x + 9 ) , h) 5 a 3 x ( 3 a + x ) ( 9 a 2 − 3 a x + x 2 ) , i) 2 a 2 b ( 2 a − 3 b ) ( 4 a 2 + 6 ab + 9 b 2 ) , j) ( a + b − 9 ) ( a − b + 1 ) , k) ( x + 3 y − 6 ) ( x − 3 y − 4 ) , l) ( x y + 1 + x − y ) ( x y + 1 − x + y ) , m) ( x y − 3 x − 2 y + 6 ) ( x y + 3 x + 2 y + 6 ) .
Explanation
Understanding the Problem We are given a list of expressions and we want to factorize them completely.
Factorizing a) a) 4 x 4 + 81 = 4 x 4 + 36 x 2 + 81 − 36 x 2 = ( 2 x 2 + 9 ) 2 − ( 6 x ) 2 = ( 2 x 2 + 6 x + 9 ) ( 2 x 2 − 6 x + 9 )
Factorizing b) b) x 4 + 625 y 4 = x 4 + 50 x 2 y 2 + 625 y 4 − 50 x 2 y 2 = ( x 2 + 25 y 2 ) 2 − ( 5 s q r t 2 x y ) 2 = ( x 2 + 5 s q r t 2 x y + 25 y 2 ) ( x 2 − 5 s q r t 2 x y + 25 y 2 )
Factorizing c) c) a 4 + a 2 b 2 + b 4 = a 4 + 2 a 2 b 2 + b 4 − a 2 b 2 = ( a 2 + b 2 ) 2 − ( ab ) 2 = ( a 2 + ab + b 2 ) ( a 2 − ab + b 2 )
Factorizing d) d) a 4 + 2 a 2 + 9 = a 4 + 6 a 2 + 9 − 4 a 2 = ( a 2 + 3 ) 2 − ( 2 a ) 2 = ( a 2 + 2 a + 3 ) ( a 2 − 2 a + 3 )
Factorizing e) e) x 4 − 39 x 2 + 25 = x 4 − 10 x 2 + 25 − 29 x 2 = ( x 2 − 5 ) 2 − ( s q r t 29 x ) 2 = ( x 2 − s q r t 29 x − 5 ) ( x 2 + s q r t 29 x − 5 )
Factorizing f) f) x 4 y 4 + x 2 y 2 + 1 = x 4 y 4 + 2 x 2 y 2 + 1 − x 2 y 2 = ( x 2 y 2 + 1 ) 2 − ( x y ) 2 = ( x 2 y 2 + x y + 1 ) ( x 2 y 2 − x y + 1 )
Factorizing g) g) 8 x 3 + 27 = ( 2 x ) 3 + 3 3 = ( 2 x + 3 ) ( 4 x 2 − 6 x + 9 )
Factorizing h) h) 135 a 6 x + 5 a 3 x 4 = 5 a 3 x ( 27 a 3 + x 3 ) = 5 a 3 x ( 3 a + x ) ( 9 a 2 − 3 a x + x 2 )
Factorizing i) i) 16 a 5 b − 54 a 2 b 4 = 2 a 2 b ( 8 a 3 − 27 b 3 ) = 2 a 2 b ( 2 a − 3 b ) ( 4 a 2 + 6 ab + 9 b 2 )
Factorizing j) j) a 2 − 8 a − 9 + 10 b − b 2 = a 2 − 8 a + 16 − 25 − ( b 2 − 10 b + 25 ) + 25 = ( a − 4 ) 2 − ( b − 5 ) 2 − 9 = ( a − 4 + b − 5 ) ( a − 4 − b + 5 ) = ( a + b − 9 ) ( a − b + 1 )
Factorizing k) k) x 2 − 10 x + 24 + 6 y − 9 y 2 = x 2 − 10 x + 25 − 1 − 9 ( y 2 − 3 2 y + 9 1 ) + 1 = ( x − 5 ) 2 − 9 ( y − 3 1 ) 2 = ( x − 5 + 3 y − 1 ) ( x − 5 − 3 y + 1 ) = ( x + 3 y − 6 ) ( x − 3 y − 4 )
Factorizing l)
( 1 − x 2 ) ( 1 − y 2 ) + 4 x y = 1 − x 2 − y 2 + x 2 y 2 + 4 x y = x 2 y 2 − x 2 − y 2 + 1 + 4 x y = ( x y ) 2 − 2 x y + 1 − x 2 + 2 x y − y 2 = ( x y + 1 ) 2 − ( x − y ) 2 = ( x y + 1 + x − y ) ( x y + 1 − x + y )
Factorizing m) m) ( x 2 − 4 ) ( y 2 − 9 ) − 24 x y = x 2 y 2 − 9 x 2 − 4 y 2 + 36 − 24 x y = x 2 y 2 − 24 x y + 144 − 9 x 2 + 24 x y − 144 − 4 y 2 + 36 = ( x y − 12 ) 2 − 9 x 2 − 4 y 2 − 108 = ( x y ) 2 − 9 x 2 − 4 y 2 − 24 x y + 36 = ( x y − 3 x − 2 y + 6 ) ( x y + 3 x + 2 y + 6 )
Examples
Factoring is a fundamental skill in algebra that allows us to simplify complex expressions and solve equations. For example, in engineering, factoring can help determine the resonant frequencies of a circuit or the stability of a structure. In economics, it can be used to analyze supply and demand curves. Factoring also plays a crucial role in cryptography, where it is used to break down complex codes and ensure secure communication. Understanding factoring techniques enhances problem-solving abilities across various disciplines.
