Remainder Theorem...1

[nycmathguy]

Section 2.3
Question 52

The Remainder Theorem

If a polynomial f(x) is divided by x − k, then the remainder is r = f(k).

1. Can you explain the Remainder Theorem statement given above?

use the Remainder Theorem
and synthetic division to find each function
value.

52. g(x) = 2x^6 + 3x^4 − x^2 + 3 at g(2).

 

[MathLover1]

The Remainder Theorem

If a polynomial f(x) is divided by x − k, then the remainder is r = f(k).


1. Can you explain the Remainder Theorem statement given above?

When we divide a polynomial f(x) by x-k the remainder is f(k), means a polynomial f(x) is not divisible by x-k, and x-k is not a factor because there is a remainder


use the Remainder Theorem
and synthetic division to find each function
value.

52. g(x) = 2x^6 + 3x^4 − x^2 + 3 at g(2).

=>the remainder is r = g(2) => k=2 and you divide polynomial by x-2

........(2x^5+4x^4+11x^3+22x^2+43x+86
(x-2)|2x^6 +0*x^5+ 3x^4+0*x^3 − x^2+0*x+ 3
.........2x^6-4x^5.........subtract
..................4x^5.........bring the next term down
..................4x^5+ 3x^4
..................4x^5-8x^4........subtract
...........................11x^4.........bring the next term down
...........................11x^4+0*x^3
...........................11x^4-22x^3......subtract
......................................22x^3........bring the next term down
......................................22x^3− x^2
......................................22x^3−44 x^2.....subtract
................................................43x^2......bring the next term down
................................................43x^2+0*x
................................................43x^2-86x........subtract
...........................................................86x.....bring the next term down
...........................................................86x+3
...........................................................86x-172.......subtract
.................................................................175->reminder

since the remainder is 175 the value of g(2) = 175

check:
g(2) = 2(2)^6 + 3(2)^4 −(2)^2 + 3
g(2) =128 + 48−4 + 3
g(2) =175

 

[nycmathguy]

The Remainder Theorem

If a polynomial f(x) is divided by x − k, then the remainder is r = f(k).


1. Can you explain the Remainder Theorem statement given above?

When we divide a polynomial f(x) by x-k the remainder is f(k), means a polynomial f(x) is not divisible by x-k, and x-k is not a factor because there is a remainder


use the Remainder Theorem
and synthetic division to find each function
value.

52. g(x) = 2x^6 + 3x^4 − x^2 + 3 at g(2).

=>the remainder is r = g(2) => k=2 and you divide polynomial by x-2

........(2x^5+4x^4+11x^3+22x^2+43x+86
(x-2)|2x^6 +0*x^5+ 3x^4+0*x^3 − x^2+0*x+ 3
.........2x^6-4x^5.........subtract
..................4x^5.........bring the next term down
..................4x^5+ 3x^4
..................4x^5-8x^4........subtract
...........................11x^4.........bring the next term down
...........................11x^4+0*x^3
...........................11x^4-22x^3......subtract
......................................22x^3........bring the next term down
......................................22x^3− x^2
......................................22x^3−44 x^2.....subtract
................................................43x^2......bring the next term down
................................................43x^2+0*x
................................................43x^2-86x........subtract
...........................................................86x.....bring the next term down
...........................................................86x+3
...........................................................86x-172.......subtract
.................................................................175->reminder

since the remainder is 175 the value of g(2) = 175

check:
g(2) = 2(2)^6 + 3(2)^4 −(2)^2 + 3
g(2) =128 + 48−4 + 3
g(2) =175

Ok. We don't stop synthetic division until the remainder is zero. Yes?

 

[MathLover1]

no, synthetic division can have the remainder too

here is synthetic division:

Write down the first coefficient without changes:

2 |2 0 3 0 -1 0 3
| 4 8 22 44 86 172
__|_________________________
|2 4 11 22 43 86 175

We have completed the table and have obtained the following resulting coefficients: 2,4,11,22,43 , 86,175.
All the coefficients except the last one are the coefficients of the quotient, the last coefficient is the remainder.
Thus, the quotient is 2x^5+4x^4 +11x^3 +22x^2 +43x +86, and the remainder is 175.

 

[nycmathguy]

no, synthetic division can have the remainder too

here is synthetic division:

Write down the first coefficient without changes:

2 |2 0 3 0 -1 0 3
| 4 8 22 44 86 172
__|_________________________
|2 4 11 22 43 86 175

We have completed the table and have obtained the following resulting coefficients: 2,4,11,22,43 , 86,175.
All the coefficients except the last one are the coefficients of the quotient, the last coefficient is the remainder.
Thus, the quotient is 2x^5+4x^4 +11x^3 +22x^2 +43x +86, and the remainder is 175.

Thank you. I have been extremely busy. Just moved into the new place. Putting math on hold for a few days. Currently in Section 2.3 and will be here until further notice. I am going nuts, literally. The air mattress is breaking my back.