Determine the Formula of Factor Theorem

Module

In

Math

(Factor Theorem)

Submitted By:

         Jean Anne Marie C. Romulo

                G10- Aguinaldo

                                                Submitted To:

                                                             Sir Nick Hermoso

Factor Theorem

1. Objectives

• Determine the formula of Factor Theorem
• Able to explain what Factor Theorem is and use this to solve a problem.

1. Topic

In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem.The factor theorem states that a polynomial [pic 1] has a factor [pic 2] if and only if [pic 3] (i.e. [pic 4] is a root).

The point of the Factor Theorem is the reverse of the Remainder Theorem: If you synthetic-divide a polynomial by x = a and get a zero remainder, then, not only is x = a a zero of the polynomial (courtesy of the Remainder Theorem), but x – a is also a factor of the polynomial (courtesy of the Factor Theorem).

It is commonly applied to factorizing and finding the roots of polynomial equations. The theorem states that [pic 5] is a factor of a polynomial f(x)if [pic 6];that is, r is a root off(x). Factor theorem problems are typically solved by applying synthetic division and then checking for a zero remainder. The remainder theorem is often used with the factor theorem. It states that if a polynomial f(x) is divided by a linear divisor [pic 7], the remainder is f(r). If [pic 8], then the remainder is 0 and [pic 9], showing that [pic 10] is a factor of f(x).

1. 5 Solving Problem

1. Show that y+4 is a factor of g(y) = 5y4+16y3−15y2+8y+16

                     g(y) = 5y4+16y3-15y2+8y+16

                   g(-4) = 5(-4)4+16(-4)3+ 15 (-4)2 + 8(-4) +16

                            = 5(256) + 16 (-64) + 15(16) + 8(-4) +16

                            = 1280 - 1024 – 240  - 32 + 16

                                = 0

1. Determine whether x-2 is a factor of x2 – 7x + 10

=x2 -7x +10

=22 – 72 + 10

= 4 – 14 + 10

= 0

1. Determine whether x-3 is the factor of x3 – 3x2 + 4x -12

= (3)3 – 3(3)2 + 4(3) – 12

= 27 – 27 + 12 -12

= 0

1. Show that x+1 is a factor of 2x3 + 5x – 9x -12

         = 2(-1)3 + 5(-1) – 9(-1) – 12

        = -2 + 5 + 9 -12

        = 0

1. Show that x + 1 is a factor of 2x – 2x2 – 2x +12

= 2(1) – 2(1)2 – 2(1) + 12

= 2 – 2 – 2 + 2

= 0

1. 10 Exercise

1. Show that (x+4) is a factor of  4x + 8x2 – 6x + - 7

1. Show that (x+1) is the factor of 3x +8x -5x +1

1. Show that (x-1) is the factor of 3x3 – 8x2  + 3x + 2

1. Show that (x-2) is the factor of x3 - 7x + 5

1. Show that (x-2) is the factor of 4x4 – 3x3 –x2 + 2x + 1

1. Show that (x-2) is the factor of 7x2 + 4x3 –x2 + 2x +1

1. Show that (x+3) is the factor of 4x2 – 2x2 - x 2 – 2x +1

1. Show that (x+3) is the factor of 2x2 – 3x2 –x2+2x + 1

1. Show that (x-4) is the factor of 2x2 – 5x – 12

1. Show that (x-5) is the factor of x3 -25x -5

V. References

http://math.tutorvista.com/algebra/factor-theorem.html?view=simple

http://www.everythingmaths.co.za/maths/grade-12/05-polynomials/05-polynomials-04.cnxmlplus 

http://math.tutorvista.com/algebra/factor-theorem.html?view=simple

http://www.niagaramathematicsacademy.com/textbook/fundamentals-of-mathematics/5-algebra-part-ii-factoring-polynomial-expressions/5-5-the-factor-theorem/

1. Key To Correction

1. =  4(4) + 8(4)2 + 6(4) -7

= 16 + 128 + 24 – 7

= 161

• X-4 is not a factor of 4x + 8x2 – 6x + - 7

1. = 3(-1) + 8(-1) – 5(-1) + 1

= -3 + -  8 - -10  + 1

= 0

1. = 3(1)3 – 8(1)2 + 3(1) +2

= 3 – 8 + 3 + 2

= 0

1. =(2)3-7(2) + 5

= -1

• x+1 is not a factor of x3 - 7x + 5

1. = 4(2)4 – 3(2)3 – (2)2 + 2(2) + 1

= 64 – 24 – 4 – + 4 + 1

= 41

• x-2 is not a factor of 4x4 – 3x3 –x2 + 2x + 1

1. = 7(2)2 – 4(2)3 –(2)2  +2(2) + 1

= 28 – 32 - -4 + 4 + 1