Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some of the factors may not be binomials.
169x^3-1690x^2-9x+90; x-10
This is an exercise in polynomial long division. Divide 169x^3 -1690x^2 -9x +90 by x-10 to get the quotient, which is 169 x^2 - 9. That happens to be the difference of the squares of 13x and 3. So it is possible to factor the quotient one more time.
If you need to review polynomial long division, see
http://www.sosmath.com/algebra/factor/fac01/fac01.html
This division turns out to be an easy one, with no remainder.
Thanks!
The complete answer is
169x^3 - 1690x^2 - 9x + 90
= (x-10)(13x + 3)(13x - 3)
To find the remaining factors of a polynomial when one factor is given, we can use polynomial division.
Step 1: Write down the given factor and the polynomial.
Given factor: x - 10
Polynomial: 169x^3 - 1690x^2 - 9x + 90
Step 2: Perform polynomial division.
To do polynomial division, divide the first term of the polynomial (169x^3) by the first term of the given factor (x). This gives you 169x^2. Write this as the first term of the quotient.
169x^2
_____________________
x - 10 | 169x^3 - 1690x^2 - 9x + 90
Now, multiply the given factor (x - 10) by 169x^2:
169x^2(x - 10) = 169x^3 - 1690x^2
Subtract this result from the polynomial:
169x^3 - 1690x^2 - 9x + 90
- (169x^3 - 1690x^2)
_______________________
- 9x + 90
Step 3: Repeat the process with the new polynomial.
Now, we have a new polynomial -9x + 90. Divide the first term (-9x) by the first term of the given factor (x). This gives you -9. Write this as the next term of the quotient.
169x^2 - 9
_____________________
x - 10 | 169x^3 - 1690x^2 - 9x + 90
- (169x^3 - 1690x^2)
_______________________
- 9x + 90
Next, multiply the given factor (x - 10) by -9:
-9(x - 10) = -9x + 90
Subtract this result from the new polynomial:
-9x + 90
- (-9x + 90)
_______________
0
Since the result is zero, there is no remainder, which means that the polynomial is completely divisible by the given factor.
Step 4: Write down the quotient.
The quotient we obtained from polynomial division is 169x^2 - 9.
Therefore, the remaining factors of the polynomial 169x^3 - 1690x^2 - 9x + 90 are:
(x - 10)(169x^2 - 9)
Note: In some cases, the quotient may not be a binomial, as it depends on the degree of the polynomial and the given factor.