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Algebra
Polynomial Examples
Factoring Polynomial Examples
Common Factors
xy + xz = x ( y + z )
In the example above, x is a common factor of xy and xz on the left side of the equation.
Greatest Common Factor (GCF)
8x
^{2}
+ 12x = 4x ( 2x + 3 )
In the example above, 4x is the GCF of the term 8x
^{2}
+ 12x
Factoring Polynomial Example 1
Question
Factoring the polynomial x
^{2}
+ 2x
Solution
x
^{2}
+ 2x = x ( x + 2 )
In the question above, x is common factor of x
^{2}
+ 2x on the left side of equation.
Factoring Polynomial Example 2
Question
Factoring the polynomial 12x
^{3}
+ 6x
^{2}
Solution
12x
^{3}
+ 6x
^{2}
= 6x
^{2}
( 2x + 1 )
In the question above, 6x
^{2}
is GCF of the term 12x
^{3}
and 6x
^{2}
on the left side of the equation.
Factoring Polynomial Example 3
Question
Factoring the polynomial x
^{2}
- 1
Solution
x
^{2}
- 1
= x
^{2}
+ x - x - 1
= x ( x + 1 ) - ( x + 1 )
= ( x + 1 ) ( x - 1 )
In the question above, (x + 1) is common factor and x
^{2}
- 1 can be factor as (x + 1)(x - 1)
Factoring Polynomial Example 4
Question
Factoring the polynomial a
^{2}
- b
^{2}
Solution
a
^{2}
- b
^{2}
= a
^{2}
+ ab - ab - b
^{2}
= a ( a + b ) - b ( a + b )
= ( a + b ) ( a - b )
In the question above, (a + b) is common factor and a
^{2}
- b
^{2}
can be factor as (a + b)(a - b)
Factoring Polynomial Example 5
Question
Factoring the polynomils: x
^{3}
- 2x
^{2}
- 5x + 6
Solution
Since 6 = 2 × 3, so try to divide ( x - 3) first
Thus, x
^{3}
- 2x
^{2}
- 5x + 6 = ( x - 3 )( x
^{2}
+ x - 2 )
Now, factoring x
^{2}
+ x - 2
Therefore, x
^{3}
- 2x
^{2}
- 5x + 6 = (x - 3)(x - 1)( x + 2)
Rule of Multiply Two Polynomials
Multiply two polynomials,
multiply each term of one polynomial by each term of the other and then simplify the result by combining like terms.
Multiply Polynomials Example 1
Question
Multiply two binomials ( x + a ) ( x + b ) = ?
Solution:
( x + a ) ( x + b )
= (x)(x) + (x)(b) + (a)(x) + (a)(b)
= (x)(x) + (b)(x) + (a)(x) + (a)(b)
= x
^{2}
+ (a + b)x + a b
note: bx and ax are similar (like) terms
Multiply Polynomials Example 2
Example
Multiply two binomials ( x + 3 ) ( x + 4 ) and write the product directly
Solution
Multiply Polynomials Example 3
Example
Multiply polynomial ( 2x + 3 ) ( 3x - 5 ) = ?
Solution
( 2x + 3 ) ( 3x - 5 )
= (2x)(3x) + (2x)(-5) + (3)(3x) + (3)(-5)
= 6x
^{2}
- 10x + 9x - 15
= 6x
^{2}
- x - 15