846.sagews

def synthetic_division(polynomial, x):
    value = 0
    quotient = []
    for coefficient in polynomial:
        value = value*x + coefficient
        quotient = quotient + [value]
    return quotient

Example 1  Long Division of Polynomials

Divide $x^2 + 10x + 21$ by $x + 3$.

x = -3
quotient = []
value = 0
for coefficient in [1, 10, 21]:
    value = value*x + coefficient
    quotient = quotient + [value]
    print quotient

[1] [1, 7] [1, 7, 0]

synthetic_division([1, 10, 21], -3)

[1, 7, 0]

http://www.wolframalpha.com/input/?i=(x^2+%2B+10x+%2B+21)%2F(x+%2B+3)

 

EXAMPLE 2  Long Division of Polynomials

Divide $4 - 5x - x^2 + 6x^3$ by $3x - 2$.

synthetic_division([2, -1/3, -5/3, 4/3], 2/3)

[2, 1, -1, 2/3]

http://www.wolframalpha.com/input/?i=(4+-+5x+-+x^2+%2B+6x^3)%2F(3x+-+2)

 

The Division Algorithm:      

$f(x) = q(x) \cdot d(x) + r(x)$

$r(x)$ will always be one less degree than $d(x)$.

$\frac{f(x)}{d(x)} = q(x) + \frac{r(x)}{d(x)}$

 

EXAMPLE 3  Long Division of Polynomials

Divide $6x^4 + 5x^3 + 3x - 5$ by $3x^2 - 2x$.

(6*x^4 + 5*x^3 + 3*x - 5)/(3*x^2 - 2*x)

http://www.wolframalpha.com/input/?i=(6x^4+%2B+5x^3+%2B+3x+-+5)%2F(3x^2+-+2x)

EXAMPLE 4  Using Synthetic Division

Use synthetic division:  $(5x^3 + 6x + 8) ÷ (x + 2)$.

synthetic_division([5, 0, 6, 8], -2)

value = 0
quotient = []
x = -2

for coefficient in [5, 0, 6, 8]:
    value = value*x + coefficient
    quotient = quotient + [value]
quotient

Remainder Theorem

Consider $\frac{f(x)}{x - c}$.

In terms of the Division Algorithm, $f(x) = q(x) \cdot (x - c) + r(x)$.

Since $r(x)$ will be one less degree than $(x - c)$, $r(x)$ is just some constant, $r$:  $f(x) = q(x) \cdot (x - c) + r$

If we evaluate $f(c)$, $f(c) = q(c) \cdot (c - c) + r$ = $r$.

Therefore, $f(c) = r$.

 

EXAMPLE 5  Using the Remainder Theorem to Evaluate a Polynomial Function

Use the Remainder Theorem to find $f(2)$ where $f(x) = x^3 - 4x^2 + 5x + 3$.

x = 2
quotient = []
value = 0

for coefficient in [1, -4, 5, 3]:
    value = value*x + coefficient
    quotient = quotient + [value]
quotient

synthetic_division([1, -4, 5, 3], 2)

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_2.py", line 7, in <module> exec compile(ur'synthetic_division([_sage_const_1 , -_sage_const_4 , _sage_const_5 , _sage_const_3 ], _sage_const_2 )' + '\n', '', 'single') File "", line 1, in <module> NameError: name 'synthetic_division' is not defined

f(x) = x^3 - 4*x^2 + 5*x + 3

f(2)

Factor Theorem

If $f(x)$ is a polynomial:

If $f(c) = 0$, then $(x - c)$ is a factor of $f(x)$, and

if $(x - c)$ is a factor of $f(x)$, then $f(c) = 0$.

EXAMPLE 6  Using the Factor Theorem

$f(x) = 2x^3 - 3x^2 - 11x + 6$.  Solve $f(x) = 0$ given that $f(3) = 0$.

x = 3
value = 0
quotient = []

for coefficient in [2, -3, -11, 6]:
    value = value*x + coefficient
    quotient = quotient + [value]
quotient

x = 1/2
value = 0
quotient = []

for coefficient in [2, 3, -2]:
    value = value*x + coefficient
    quotient = quotient + [value]
quotient

f(x) = 2*x^3 - 3*x^2 - 11*x + 6

factor(f)

solve(f == 0, x)

plot(f, -3, 4) + points([(-2, 0), (1/2, 0), (3, 0)], color = 'red')