# Use of Factor Theorem

**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.- 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.
**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).**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.

## Exercise

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

(v) k (x -3), k 0

(vi) x

(vii) 4 x - Let f(x) = 3 x³ -7 x² +1. Find the remainder when f(x) is divided by

(i) x +2

(ii) 2 x +2

(iii) 1(x +2)/2

(iv) 3(x +2)

(v) k (x +2), k 0

(vi) x

(vii) 10 x - 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.
- Show that x -1 is a factor of 2 x² +x -3. Hence factorise 2 x² +x -3 completely.
- Show that 2 x +3 is a factor of 6 x² +5 x -6. Hence find the other factor.
- Show that x +2 is a factor of f(x) = x³ +2 x² -x -2. Hence factorise f(x) completely.
- Show that x -1 is a factor of x
^{5}-1 while x^{5}+1 is not divisible by x -1. - 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.
- The 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.
- 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.
- If x² -1 is a factor of f(x) = x
^{4}+ax +b, calculate the values of a and b. Using these values of a and b, factorise f(x). - 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.
- The polynomial x
^{4}+bx³ +59 x² +cx +60 is exactly divisible by x² +4 x +3. Find the values of b and c.

## Answers

**1.**Dividend = 2 x³ -7 x² +5 x +9, divisor = 2 x -3

quotient = x² -2 x -1/2, remainder = -21/2

**2.**-21/2

**3.**(i) 38 (ii) -176

**4.**(i) -4 (ii) - 7 (iii) 29 (iv) -4 (v) -4 (vi) -1 (vii) -1

**5.**(i) -51 (ii) -9 (iii) -51 (iv) -51 (v) -51 (vi) 1 (vii) 1

**6.**4

**7.**(x -1)(2 x +3)

**8.**3 x -2

**9.**(x -1)(x +1)(x +2)

**11.**k = -4; (2 x -1)²

**12.**a = 3, b = -3; (x +2)(2 x +1)(x -1)

**13.**a = -1/2, b = 0, x (x -1)(x +1)

**14.**a = 0, b = -1, (x -1)(x +1)(x² +1)

**15.**a = -6, b = -8, third factor is x +4

**16.**b = 13, c = 107