6.3 D1
February 8, 2016
Unit 5
Unit 5 ‑ Polynomials and Polynomial Functions 6.3 Dividing Polynomials
Objective:
You will be able to divide polynomials using synthetic division.
You will be able to factor a polynomial completely using division.
Bell Ringer
Write each polynomial in standard form. Then list the
coefficients.
1.) 5x - 2x2 + 9 + 4x3
2.) 10 + 5x3 - 9x2
3.) 3x + x2 - x + 7 - 2x2
4.) -4x4 - 7x2 + x3 + x4
1
6.3 D1
Synthetic Division
Steps:
1.) Set the divisor equal to zero and solve for the variable.
2.) List in only one row the value found for the variable first and then all
of the coefficients of the polynomial in descending order by degree. If
a degree is skipped the coefficient is 0.
3.) Place a half box around the first number in the row.
4.) Skip a line (leave a space) and then draw a line that extends the
length of the row.
5.) Bring the first coefficient down below the line. (Rewrite the number
in the same column below the line.)
6.) Multiply the number below the line by the number in the half box.
7.) Write this product above the line under the next coefficient.
8.) Add the numbers in the column. Write the sum below the line in the
same column.
9.) Multiply the sum found in step 8 by the number in the box.
10.) Repeat steps 7 - 9 for the remaining coefficients.
11.) The last sum you find is the remainder of the polynomial
12.) Write the quotient in the form of a polynomial. Each number below
the line is a coefficient of the polynomial quotient.
Begin with the first number below the line. This is the coefficient
of the first term in the quotient. The degree of its variable is one
less than the degree of the polynomial that was divided.
The power of the variables for the remaining coefficients are in
descending order working from left to right.
The last sum you find is the remainder of the polynomial and is
not given a variable.
Dividing Polynomials
Synthetic Division:
A method of dividing polynomials by a linear divisor.
Example: Use synthetic division to divide
3x3 - 4x2 + 2x - 1 by x + 1.
Above the line - add
Below the line - multiply
2
6.3 D1
Dividing Polynomials
Determining if a binomial is a factor of a polynomial.
If the remainder IS equal to zero then the divisor is a
factor of the dividend.
(The binomial IS a factor of the polynomial.)
If the remainder is NOT equal to zero then the divisor is
not a factor of the dividend.
(The binomial is NOT a factor of the polynomial.)
Synthetic Division
Divide the polynomial. State whether the divisor is a
factor of the dividend.
9x3 - 18x2 + x + 2
3x + 1
3
6.3 D1
Synthetic Division
Divide the polynomial. State whether the divisor is a
factor of the dividend.
x3 + 27
x+3
Synthetic Division
You Try
Divide the following polynomials. State whether each
divisor is a factor of the dividend.
1.) 2x3 - 3x2 - 18x - 8
2.) x5 - 3x3 - 4x - 1
x-4
x-1
4
6.3 D1
Real Wold Application
The volume in cubic feet of a sarophagus (excluding the cover)
can be expressed as the product of its three dimensions:
V(x) = x3 - 13x2 + 12. The length is x + 4. Find linear
expressions with integer coefficients for the other dimensions.
Assume that the width is greater than the height.
Synthetic Division
Use the given factor to factor they polynomial
completely.
y = x3 - 13x - 12; (x - 4)
5
6.3 D1
Synthetic Division
You Try
Use the given factor to factor they polynomial
completely.
y = x3 + 4x2 + x - 6; (x + 3)
Homework
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