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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

Thus, x³ - 2x² - 5x + 6 = ( x - 3 )( x² + x - 2 )

Now, factoring x² + x - 2

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

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