Hello,
let's dividing f(x) by x - 7, using syntethic division:
7 | 1.... 0.. - 47.. - 16.... 8.... 52
(carrying 1 down)
-------- -------- ----------- --------- --
.... 1
(multiplying 1 by 7 and writing the result beneath 0)
...... .... 7
(adding 0 and 7 together)
.... 1.... 7
(multiplying 7 by 7 and writing the result beneath - 47)
...... .... 7.... 49
(adding - 47 and 49 together)
.... 1.... 7..... 2
(multiplying 2 by 7 and writing the result beneath - 16)
...... .... 7.... 49.... 14
(adding - 16 and 14 together)
.... 1.... 7..... 2.... - 2
(multiplying - 2 by 7 and writing the result beneath 8)
7 | 1.... 0.. - 47.. - 16..... 8.... 52
...... .... 7.... 49.... 14. - 14
(adding 8 and - 14 together)
.... 1.... 7..... 2.... - 2... - 6
(multiplying - 6 by 7 and writing the result beneath 52)
...... .... 7.... 49.... 14. - 14.. - 42
(adding 52 and - 42 together, obtaining the remainider)
.... 1.... 7..... 2.... - 2... - 6..... 10 (remainder)
being 10 the remainder of the division, by the remainder theorem the value of f(x) for x = 7 is just 10
in fact, being the quotient (1)x⁴ + (7)x³ + 2x² + (- 2)x + (- 6)x = x⁴ + 7x³ + 2x² -
2x - 6 and the remainder 10, the result of the division can be expressed as:
x⁵ - 47x³ - 16x² + 8x + 52 = (x - 7)(x⁴ + 7x³ + 2x² - 2x - 6) + 10 =
(letting x = 7)
[(7) - 7] [(7)⁴ + 7(7)³ + 2(7)² - 2(7) - 6] + 10 =
0 ∙ [(7)⁴ + 7(7)³ + 2(7)² - 2(7) - 6] + 10 =
0 + 10 =
10
I hope it's helpful
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Answers & Comments
Verified answer
Hello,
let's dividing f(x) by x - 7, using syntethic division:
7 | 1.... 0.. - 47.. - 16.... 8.... 52
(carrying 1 down)
7 | 1.... 0.. - 47.. - 16.... 8.... 52
-------- -------- ----------- --------- --
.... 1
(multiplying 1 by 7 and writing the result beneath 0)
7 | 1.... 0.. - 47.. - 16.... 8.... 52
...... .... 7
-------- -------- ----------- --------- --
.... 1
(adding 0 and 7 together)
7 | 1.... 0.. - 47.. - 16.... 8.... 52
...... .... 7
-------- -------- ----------- --------- --
.... 1.... 7
(multiplying 7 by 7 and writing the result beneath - 47)
7 | 1.... 0.. - 47.. - 16.... 8.... 52
...... .... 7.... 49
-------- -------- ----------- --------- --
.... 1.... 7
(adding - 47 and 49 together)
7 | 1.... 0.. - 47.. - 16.... 8.... 52
...... .... 7.... 49
-------- -------- ----------- --------- --
.... 1.... 7..... 2
(multiplying 2 by 7 and writing the result beneath - 16)
7 | 1.... 0.. - 47.. - 16.... 8.... 52
...... .... 7.... 49.... 14
-------- -------- ----------- --------- --
.... 1.... 7..... 2
(adding - 16 and 14 together)
7 | 1.... 0.. - 47.. - 16.... 8.... 52
...... .... 7.... 49.... 14
-------- -------- ----------- --------- --
.... 1.... 7..... 2.... - 2
(multiplying - 2 by 7 and writing the result beneath 8)
7 | 1.... 0.. - 47.. - 16..... 8.... 52
...... .... 7.... 49.... 14. - 14
-------- -------- ----------- --------- --
.... 1.... 7..... 2.... - 2
(adding 8 and - 14 together)
7 | 1.... 0.. - 47.. - 16..... 8.... 52
...... .... 7.... 49.... 14. - 14
-------- -------- ----------- --------- --
.... 1.... 7..... 2.... - 2... - 6
(multiplying - 6 by 7 and writing the result beneath 52)
7 | 1.... 0.. - 47.. - 16..... 8.... 52
...... .... 7.... 49.... 14. - 14.. - 42
-------- -------- ----------- --------- --
.... 1.... 7..... 2.... - 2... - 6
(adding 52 and - 42 together, obtaining the remainider)
7 | 1.... 0.. - 47.. - 16..... 8.... 52
...... .... 7.... 49.... 14. - 14.. - 42
-------- -------- ----------- --------- --
.... 1.... 7..... 2.... - 2... - 6..... 10 (remainder)
being 10 the remainder of the division, by the remainder theorem the value of f(x) for x = 7 is just 10
in fact, being the quotient (1)x⁴ + (7)x³ + 2x² + (- 2)x + (- 6)x = x⁴ + 7x³ + 2x² -
2x - 6 and the remainder 10, the result of the division can be expressed as:
x⁵ - 47x³ - 16x² + 8x + 52 = (x - 7)(x⁴ + 7x³ + 2x² - 2x - 6) + 10 =
(letting x = 7)
[(7) - 7] [(7)⁴ + 7(7)³ + 2(7)² - 2(7) - 6] + 10 =
0 ∙ [(7)⁴ + 7(7)³ + 2(7)² - 2(7) - 6] + 10 =
0 + 10 =
10
I hope it's helpful