Divide a polynomial of degree 3 or 4 by a linear polynomial using different methods of division and check whether the answers are unique
Answers
Answer:
Let’s first get the problem set up.
x
−
4
5
x
3
−
x
2
+
0
x
+
6
Recall that we need to have the terms written down with the exponents in decreasing order and to make sure we don’t make any mistakes we add in any missing terms with a zero coefficient.
Now we ask ourselves what we need to multiply
x
−
4
to get the first term in first polynomial. In this case that is
5
x
2
. So multiply
x
−
4
by
5
x
2
and subtract the results from the first polynomial.
5
x
2
x
−
4
5
x
3
−
x
2
+
0
x
+
6
−
(
5
x
3
−
20
x
2
)
––––––––––––––
19
x
2
+
0
x
+
6
The new polynomial is called the remainder. We continue the process until the degree of the remainder is less than the degree of the divisor, which is
x
−
4
in this case. So, we need to continue until the degree of the remainder is less than 1.
Recall that the degree of a polynomial is the highest exponent in the polynomial. Also, recall that a constant is thought of as a polynomial of degree zero. Therefore, we’ll need to continue until we get a constant in this case.
Here is the rest of the work for this example.
5
x
2
+
19
x
+
76
x
−
4
5
x
3
−
x
2
+
0
x
+
6
−
(
5
x
3
−
20
x
2
)
––––––––––––––
19
x
2
+
0
x
+
6
−
(
19
x
2
−
76
x
)
––––––––––––––
76
x
+
6
−
(
76
x
−
304
)
–––––––––––––
310
Okay, now that we’ve gotten this done, let’s remember how we write the actual answer down. The answer is,
5
x
3
−
x
2
+
6
x
−
4
=
5
x
2
+
19
x
+
76
+
310
x
−
4