9th Use of Factor Theorem

Use of Factor Theorem IX
Division Algorithm for Polynomials:

If a polynomial f(x) is divided by a non-zero polynomial g (x) then there exist unique polynomials q (x) and r (x) such that f(x)=g(x)q(x) + r(x) where either r (x) = 0 or deg r (x) < deg g(x).Here dividend = f(x), divisor = g (x), quotient = q (x) and remainder = r (x). Thus if g (x) is a quadratic polynomial, then remainder r (x) is of the form ax+b, where a, b may be zero. If divisor g (x) is a linear polynomial then r (x) is a constant polynomial, i.e., r (x) = c, where c may be zero.

1. A non-zero polynomial g (x) is called a factor of any polynomial f (x) iff there exists some polynomial q (x) such that f(x) = g(x)q(x), i.e. iff on dividing f(x) by g (x), the remainder is zero.
2. Remainder Theorem:
   If a polynomial f(x) is divided by (x -a), then remainder = f(a).
   If a polynomial f(x) is divided by (x +a), then remainder = f(-a).
   If a polynomial f(x) is divided by (ax +b), then remainder = f(-b/a).
3. Factor Theorem:
   If f(x) is a polynomial and a is a real number, then (x -a) is a factor of f(x) iff f(a)=0.

Test Paper 

1.Divide 2 x³ -7 x² +5 x +9 by 2 x -3 by long division method. Mention the dividend, divisor, quotient and remainder.
2. Using remainder theorem, find the remainder when 2 x³ -7 x² +5 x +9 is divided by 2 x -3. 

3. Find the remainder (without division) on dividing f(x) by x +3, where

(i) f(x) = 2 x² -7 x -1 (ii) f(x) = 3 x³ -7 x² +11 x +1. 

4. Let f(x) = 2 x² -7 x -1. Find the remainder when f(x) is divided by

(i) x -3 (ii) 2x -3 (iii) x /2 -3 (iv) 2(x -3)

5. The polynomial x4 +bx³ +59 x² +cx +60 is exactly divisible by x² +4 x +3. Find the values of b and c.

6. Using remainder theorem, find the value of a if the division of x³ +5 x² -ax +6 by (x -1) leaves the remainder 2 a.

7. Show that x -1 is a factor of 2 x² +x -3. Hence factorise 2 x² +x -3 completely.

8. Show that 2 x +3 is a factor of 6 x² +5 x -6. Hence find the other factor.

9.Show that x +2 is a factor of f(x) = x³ +2 x² -x -2. Hence factorise f(x) completely. 

10. Show that x -1 is a factor of x5 -1 while x5 +1 is not divisible by x -1.

11. Find the constant k if 2 x -1 is a factor of f(x) = 4 x² +kx +1. Using this value of k, factorise f(x) completely.

12The expression 2 x³ +a x² +bx -2 leaves remainders of 7 and 0 when divided by 2 x -3 and x +2 respectively. Calculate the values of a and b. With these values of a and b, factorise the expression completely.

13.If x +1 and x -1 are factors of f(x) = x³ +2 ax +b, calculate the values of a and b. Using these values of a and b, factorise f(x) completely. 

14. If x² -1 is a factor of f(x) = x4 +ax +b, calculate the values of a and b. Using these values of a and b, factorise f(x). 

15..Given that x² -x -2 is a factor of x³ +3 x² +ax +b, calculate the values of a and b and hence find the remaining factor.