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Dividing Polynomials MR. VELAZQUEZ
HONORS PRECALCULUS
Long Division of Polynomials
Long Division of Polynomials
Long Division of Polynomials
2
3 2 9 6 5x x x
12 5x
4
13
3x
29 6x x
12 8x
13
3 2x
Long Division of Polynomials
2 3
3 2
2
2
x +5x -3 x 3x 2
x +5x 3x
-5x 6x 2
-5x 25x 15
31x- 17
5x 2
31 17
5 3
x
x x
You need to leave a hole when you have missing terms. This technique will help you line up like terms. See the dividend in this example.
The Division Algorithm
Examples Divide using Long Division.
3 22 5 6 4 +7x x x
Examples Divide using Long Division.
2 4 32 1 8 3 +5 1x x x x x
Exit Ticket, Part 1 (on separate sheet) Divide the following polynomials using long division.
𝑥3 + 7𝑥2 + 17𝑥 + 20
𝑥 + 4
𝑥4 − 3𝑥2 + 𝑥 − 4
𝑥 − 2 1. 2.
Synthetic Division
Comparing Methods
Remember to use —2 for the divisor, and put in a 0 for the missing term.
Dividing 5𝑥3 + 6𝑥 + 8 by 𝑥 + 2
Synthetic Division
2 5 + 7 - 1
Notice that the divisor has to be a binomial of degree 1 with no coefficients.
5
10
3
6
5
2
55 3
2
2 5 7 1
xx
x x x
Thus:
Examples
Divide using synthetic division.
3 23 5 7 8
4
x x x
x
The Remainder Theorem
If you are given the function f(x)=x3- 4x2+5x+3 and you want to find f(2), then the remainder of this function when divided by x-2 will give you f(2)
f(2)=5
The Remainder Theorem
2(1) for f(x)=6x 2 5 is
1 6 -2 5
6 4
6 4 9
f(1)=9
f x
Examples
Use synthetic division and the remainder theorem to find the indicated function value.
3 2( ) 3 5 1; f(2)f x x x
The Factor Theorem
Solve the equation 2x3-3x2-11x+6=0 given that 3 is a zero of f(x)=2x3-3x2-11x+6. The factor theorem tells us that x-3 is a factor of f(x). So we will use both synthetic division and long division to show this and to find another factor.
Another factor
Examples Solve the equation 5x2 + 9x – 2=0 given that -2 is a zero of f(x)= 5x2 + 9x - 2
Examples Solve the equation x3- 5x2 + 9x - 45 = 0 given that 5 is a zero of f(x)= x3- 5x2 + 9x – 45. Consider all complex number solutions.
Exit Ticket, Part 2 (same paper as Pt. 1) Use synthetic division to find the following:
𝑥3 + 𝑥2 + 2𝑥 − 8
𝑥 − 3 3. 𝑥4 + 5𝑥 − 12
𝑥 + 2 4. Find 𝑓(2) of the function 𝑓 𝑥 =
𝑥3 + 𝑥2 − 11𝑥 + 10, and show that it is equal to the remainder when dividing 𝑓(𝑥) by 𝑥 − 2.
5.
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