Download Division of Polynomials

Division of Polynomials
The Long and Short of it.
Review of Vocabulary
 Factor
 root, zero, solution, x-intercept
 dividend ÷ divisor = quotient
 Example: in 12 ÷ 3 = 4:
12 is the dividend
3 is the divisor
4 is the quotient
 remainder
Polynomial Long Division
• If you're dividing a polynomial by
something more complicated than just
a simple monomial, then you'll need to
use a different method for the
simplification. That method is called
"long (polynomial) division", and it
works just like the long (numerical)
division you did back in elementary
school, except that now you're
dividing with variables.
Dividing by a binomial
• The next few slides demonstrate how
to do the long division. The slides are
meant as background information.
You will find the same steps at :
Purplemath
I will demonstrate a quicker way also.
Example 1
• Divide x2 – 9x – 10 by x + 1
• The process is similar to long division of
numbers as you learned in elementary
school.
First, I set up the division: For the
moment, I'll ignore the other terms and
look just at the leading x of the divisor and
the leading x2 of the dividend.
If I divide the leading x2 inside by the
leading x in front, what would I get? I'd get
an x. So I'll put an x on top:
Now I'll take that x, and multiply it through
the divisor, x + 1. First, I multiply the x (on
top) by the x (on the "side"), and carry the
x2 underneath:
Then I'll multiply the x (on top) by the 1
(on the "side"), and carry the 1x
underneath:
Then I'll draw the "equals" bar, so I can do
the subtraction. To subtract the polynomials, I
change all the signs in the second line...
...and then I add down. The first term (the x2)
will cancel out:
I need to remember to carry down that last
term, the "subtract ten", from the dividend:
Now I look at the x from the divisor and the
new leading term, the –10x, in the bottom line
of the division. If I divide the –10x by the x, I
would end up with a –10, so I'll put that on
top:
Now I'll multiply the –10 (on top) by the
leading x (on the "side"), and carry the
–10x to the bottom:
...and I'll multiply the –10 (on top) by
the 1 (on the "side"), and carry the –10
to the bottom:
I draw the equals bar, and change
the signs on all the terms in the
bottom row:
Then I add down:
finally
• The answer is x-10.
• That means x2 – 9x – 10 divide by x + 1
= x-10.
In another words:
x2 – 9x – 10 = (x + 1)(x-10)
One more way to say: x-10 is a factor of
x2 – 9x – 10
Why is division useful?
When solving a quadratic such as 𝑥 2 −
9𝑥 − 10 = 0
You may factor the quadratic first and
then find the roots of each linear factor.
𝑥 2 − 9𝑥 − 10 = 0
(𝑥 − 1)(𝑥 + 10) = 0
𝑋 − 1 = 0 therefore 𝑥 = 1
Or 𝑥 + 10 = 0 therefore 𝑥 = −10
As useful as long division is, most people
find the steps difficult to follow.
A quicker way to divide is to use synthetic
division (AKA synthetic substitution)
The next few slide will walk you through the
steps. Please take careful notes.
Before you start to use Synthetic
Division, remember:
• 1. Both dividend and divisor have to
be in the standard form.
• 2. If a term is missing in dividend,
replace it with a zero
• 3. The leading co-efficient of the
divisor has to be 1.
Step 1
• DIVIDEND: Strip off all of the variables
and consider only coefficients.
Remember to insert zero for each
missing term.
• Find the zero of the divisor.
Bring the leading coefficient of the
dividend down. Then there is a series of
multiplication and addition.
The last term is the remainder.
If remainder (last term) is zero, you have
found a factor.
Example 1
• (x³ +16x² +59x −6) ÷ (x+6)
OR
𝑥 3 +16𝑥 2 +59𝑥 −6
𝑥+6
The dividend is: 𝒙𝟑 +𝟏𝟔𝒙𝟐 + 𝟓𝟗𝒙 − 𝟔;
consider only it coefficients:
1 16 59 -6
The divisor is 𝒙 + 𝟔; its zero is -6 since
−6+6=0
Opposite
of number in
divisor
-6
1
1
16
59
-6
-60
6
10
-1
0
x2 + 10x - 1
-6
R=0
Therefore
(x³ +16x² +59x −6) ÷ (x+6)= 𝑥2 + 10𝑥 − 1
Therefore in order to find all the zeros of
𝑥 3 + 16𝑥 2 + 59𝑥 − 6 ; one would find the
roots of the linear function x+6 and the
roots of the quadratic 𝒙𝟐 + 𝟏𝟎𝒙 − 𝟏
The roots of 𝑥 3 − 16𝑥 2 + 59𝑥 − 6 are:
-6, 0.1 and -10.1
Example 2
• (4x³ +18x² +12x −18) ÷ (x+3)
The dividend is: 4x³ +18x² +12x −18;
consider only it coefficients:
4 18 12 -18
The divisor is 𝒙 + 𝟑; its zero is -3 since
−3+3=0
Opposite
of number in
divisor
-3
4
18
-12
4
6
12
-18
-18
18
-6
0
4x2 + 6x - 6
R=0
Synthetic division and
substitution
• Synthetic division is also referred to as
synthetic substitution, because the
remainder is value of the polynomial if
the zero of the divisor is substituted in
for the variable.
Example 3: (x³ +6x² −15x −52) ÷ (x+7)
Opposite
of number in
divisor
-7
1
6
-7
1
-15
-52
7
56
-1
𝑥2 − 𝑥 − 8
-8
4
R=4
Ex 3 Continued:
Now subsitute x= -7 into
x³ +6x² −15x −52
(−7)³ +6∗(−7)² −15∗(−7)−52
Like magic, answer is 4 which is the
remainder.
• Evaluating polynomials becomes
simpler using this method, which
usually does not require a calculator.
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