Download FIB notes Remainder and Factor Theorems

FIB notes Remainder and Factor Theorems
1. The degree of a polynomial indicates how many _____________ zeros the function has.
2. If the degree of a polynomial is 6 and it only crosses the x-axis 2 times then there must be ____ imaginary zeros.
3. If a polynomial is divided by a linear expression and the remainder is zero, then the linear expression was a
_________ of the original polynomial.
4. If a polynomial is divided by (x+8) then synthetic division would use a numerical value of ____ in the box.
5. If a polynomial has a remainder of zero then the numerical value in the box was a _____ _____ of the original
equation.
6. If a polynomial is divided by (x-3) then the numerical value in the box is ____. If the remainder is zero, then
complete the following statements
3 is a
(x-3)is a
f(3)=
A point on the graph of the original polynomial is
The value of the function at 3 is
The original polynomial crosses the x-axis at
The original polynomial can be rewritten as a product of (
) and the depressed polynomial
7. When a polynomial is divided synthetically by (x-7) and the remainder is 5. Complete the following statements
7 is ______a
(x-7)is _______a
f(7)=
A point on the graph of the original polynomial is
The value of the function at 7 is
The original polynomial does __________cross the x-axis at
8. A depressed polynomial has a degree that is _________ _____________ than the original.
9. If the original polynomial has a degree of 5, then the resulting depressed polynomial after synthetically dividing
by (x+8) would have a degree of _________.
10. When synthetically dividing, if the original polynomial has a degree of 4, then you will have _____columns of
numbers out to the right side of the box. The last column is the ______________ column.
11. The analysis of the remainder is applying the Remainder Theorem! The Factor Theorem is being applied when
you make a statement about the factor after checking to see if the remainder=0. If it was zero, then is WAS a
factor. If it was NOT zero, then it was NOT a factor.
Operations with polynomials (adding and subtracting)
12. When 2 or more polynomials are added together, the sum will always be a ____________________.
13. When polynomials are added together the polynomial with the highest degree will have the ______ degree as
the answer. 7 x 3 + 9 x + 8 has a degree of _____
5 x 4 − 2 x 2 + x − 1 has a degree of _____
The sum of these is ________________________________________________ which has a degree of _____
14. When 2 polynomials are subtracted, the difference will always be a ____________________.
15. When 2 polynomials are subtracted, a negative sign must be distributed to all the terms in the second
polynomial
10 x 3 + x + 2 minus 7 x 3 − 4 x 2
Write using parentheses and then show the distribution
16. The exponents help determine “like terms” and only like terms can be added or subtracted.
The exponents DO NOT CHANGE! Only the _______________ are added or subtracted. The best way to list
answers after combining like terms is in ____________________ power order.
Operations with polynomials (multiply and divide)
17. Parentheses are very important. Look at the difference in these 2 examples
6 x + 8( x + 2)
and
(6 x + 8)( x + 2)
18. When multiplying, the exponents of the variables change.
When multiplying, the exponents are ___________
When dividing, the exponents are ___________
x3 x5
xx
xx 5
x 17
x
x7
x 10
x 4 x −5
Is this last example multiplying or dividing?
19. Multiply
(2 x − 5)( x 2 − 7 x + 1)
20. Naming
A single term polynomial is also known as a __________________
A double term polynomial is also known as a _________________
A three term polynomial is also known as a __________________
21. The prefix “poly” means __________