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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)
8x2 + 12x = 4x ( 2x + 3 )
In the example above, 4x is the GCF of the term 8x2 + 12x

Factoring Polynomial Example 1

Question
Factoring the polynomial x2 + 2x
Solution
x2 + 2x = x ( x + 2 )
In the question above, x is common factor of x2 + 2x on the left side of equation.

Factoring Polynomial Example 2

Question
Factoring the polynomial 12x3 + 6x2
Solution
12x3 + 6x2 = 6x2 ( 2x + 1 )
In the question above, 6x2 is GCF of the term 12x3 and 6x2 on the left side of the equation.

Factoring Polynomial Example 3

Question
Factoring the polynomial x2 - 1
Solution
x2 - 1
= x2 + x - x - 1
= x ( x + 1 ) - ( x + 1 )
= ( x + 1 ) ( x - 1 )
In the question above, (x + 1) is common factor and x2 - 1 can be factor as (x + 1)(x - 1)

Factoring Polynomial Example 4

Question
Factoring the polynomial a2 - b2
Solution
a2 - b2
= a2 + ab - ab - b2
= a ( a + b ) - b ( a + b )
= ( a + b ) ( a - b )
In the question above, (a + b) is common factor and a2 - b2 can be factor as (a + b)(a - b)

Factoring Polynomial Example 5

Question
Factoring the polynomils: x3 - 2x2 - 5x + 6
Solution
Since 6 = 2 × 3, so try to divide ( x - 3) first
Factoring Polynomils
Thus, x3 - 2x2 - 5x + 6 = ( x - 3 )( x2 + x - 2 )
Now, factoring x2 + x - 2
Factoring Polynomils
Therefore, x3 - 2x2 - 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)
= x2 + (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
polynomial multiply example2 solution figure

Multiply Polynomials Example 3

Example
Multiply polynomial ( 2x + 3 ) ( 3x - 5 ) = ?
Solution
( 2x + 3 ) ( 3x - 5 )
= (2x)(3x) + (2x)(-5) + (3)(3x) + (3)(-5)
= 6x2 - 10x + 9x - 15
= 6x2 - x - 15