## Problems on algebra-IV

In this page 'Problems on algebra-IV' we are going to see problems on factor theorem.

Parents and teachers can guide the students to do the problems on their own. If they are having any doubt they can verify the solutions.

Before going to the problems let us recall about the definitions of  'Remainder theorem' and  "Factor theorem".

Remainder theorem

Let p(x) be any polynomial and a be any real number.  If p(x) is divided by the linear polynomial x-a, then the remainder is p(a).

Note:

1. If p(x) is divided by (x+a), then the remainder is p(-a).
2. If p(x) is divided by (ax-b), then the remainder is p(b/a).
3. If p(x) is divided by (ax+b), then the remainder is p(-b/a).
4. Here -a, b/a, and -b/a are the zeros of the divisors x+a, ax-b and ax+b respectively.

Factor theorem

Let p(x) be a polynomial and a be any real number. If p(a) = 0, then (x-a) is a factor of p(x).

Note:

If (x-a) is a factor of p(x), then p(a)= 0.

### Problems on algebra-IV

The following problems are based on factor theorem.

1.          Determine whether (x+1) is a factor of the following polynomials.

(i)      6x + 7x³ - 5x - 4

(ii)     2x + 9x³ + 2x² + 10x + 15

(iii)    3x³ + 8x² - 6x - 5

(iv)     x³ -  14x² + 3x + 12

2.          Determine whether (x+4) is a factor of x³ + 3x² - 5x + 36.

3.          Using factor theorem show that (x-1) is a factor of

4x³ - 6x² + 9x - 7.

4.         Determine whether (2x+1) is a factor of 4x³ + 4x² - x - 1.

5.         Determine the value of p if (x+3) is a factor of x³ - 3x² - px + 24.

Students can try to solve the problems on their own. Parents and teachers can encourage the students to do so. If they are having any doubt they can verify the solutions.  If you are having any further doubt you can contact us through mail, we will help you to clear your doubt.