Question

2.) The polynomial, p (x) = 3x4 + x3 – 17x2 + 19x – 6 is divided by
d (x) = x2 – 2x + 1.
(a) Determine the quotient, q (x) and the remainder, r (x) of the polynomial,
p (x). [8]
(b) State whether or not that d (x) is a factor of p (x). Why this is so? [2]
(c) Hence, factorize p (x) completely giving the solution as a product of linear
factors. [5]

Answer 1

Answer:

Answer is in attachment

Answer 2

Given: $p(x) = 3x^{4} + x^{3} - 17x^{2} + 19x - 6$ is divided by  $d(x) = x^{2} - 2x + 1$.

To find:

(a) Quotient and remainder

(b) If d(x) is a factor

(c) Factorize p(x) completely.

Solution:

(a)

• The quotient of the quadratic equation provided in the question is as follows,

        $q(x) = 3x^{3} + 7x - 6$

• The remainder of the quadratic equation provided in the question is as follows,

        $r(x) = -10x$

(b)

• d(x) is not a factor of p(x) because d(x) does not factorize p(x) completely and hence, it leaves a remainder.

(c)

• When we factorize p(x) completely, we get the solution as,

        $(x - 1)^{2} (x + 3) (3x - 2)$

Therefore,

(a) q(x) = 3x³ + 7x - 6, r(x) = -10x

(b) d(x) is not a factor of p(x) since d(x) does not factorize p(x) completely.

(c) p(x) = (x - 1)² (x + 3) (3x - 2).