Question

Check whether each of the polynomials listed below is a factor of 3x³ − 2x² − 3x + 2; if not, find the remainder.

(i) x − 1

(ii) 3x − 2

(iii) 2x − 3

(iv) x + 1

(v) 3x + 2

(vi) 2x + 3

Answer 1

Factor theorem says that for the polynomial p(x) and for the number a, if we have p(a) = 0 then (x − a) is a factor of p(x).

Given: Polynomial 3x³ − 2x² − 3x + 2

(i)

Divisor = (x − 1)

Putting x = 1 in the given polynomial:

3(1)³ − 2(1)² − 3(1) + 2

= 3 − 2 − 3 + 2

= 0

∴(x − 1) is a factor of the polynomial 3x³ − 2x² − 3x + 2.

(ii)

Divisor = (3x − 2)

To check whether (3x − 2) is a factor of 3x³ − 2x² − 3x + 2, we have to convert (3x −2) in a form that is suitable for the application of the Factor theorem.

(3x −2) = 3

Putting x = in the given polynomial:

3 − 2 − 3 + 2

Thus, is a factor of the given polynomial.

∴(3x −2) is a factor of the polynomial 3x³ − 2x² − 3x + 2.

(iii)

Divisor = (2x − 3)

To check whether (2x − 3) is a factor of 3x³ − 2x² − 3x + 2, we have to convert (2x − 3) in a form that is suitable for the application of the Factor theorem.

(2x − 3) = 2

Putting x = in the given polynomial:

3 − 2 − 3 + 2

=

=

=

Thus, is a factor of the given polynomial.

∴(2x − 3) is not a factor of the polynomial 3x³ − 2x² − 3x + 2.

(iv)

Divisor = (x + 1)

To check whether (x + 1) is a factor of 3x³ − 2x² − 3x + 2, we have to convert (x + 1) in a form that is suitable for the application of the Factor theorem.

(x + 1) = {x − (−1)}

Putting x = −1 in the given polynomial:

3(−1)³ − 2(−1)² − 3(−1) + 2

= −3 − 2 + 3 + 2

= 0

∴(x + 1) is a factor of the polynomial 3x³ − 2x² − 3x + 2.

(v)

Divisor = (3x + 2)

To check whether (3x + 2) is a factor of 3x³ − 2x² − 3x + 2, we have to convert (3x + 2) in a form that is suitable for the application of the Factor theorem.

(3x + 2) = 3

= 3

Putting x = in the given polynomial:

3 − 2 − 3 + 2

=

=

=

_{Thus, is a factor of the given polynomial.}

∴(3x + 2) is not a factor of the expression 3x³ − 2x² − 3x + 2.

(vi)

Divisor = (2x + 3)

To check whether (2x + 3) is a factor of 3x³ − 2x² − 3x + 2, we have to convert (2x + 3) in a form that is suitable for the application of the Factor theorem.

(2x + 3) = 2

= 2

Putting x = in the given polynomial:

3 − 2 − 3 + 2

=

=

=

_{Thus, is not a factor of the given polynomial.}

∴(2x + 3) is not a factor of the polynomial 3x³ − 2x² − 3x + 2.