Question

While factorizing a given polynomial, using remainder and factor theorem, a student finds that $$(2x + 1)$$ is a factor of $$2x^{3} + 7x^{2} + 2x - 3$$. 

1. Is the student’s solution correct stating that $$(2x + 1)$$ is a factor of the given polynomial? 
2. Give a valid reason for your answer. Also, factorize the given polynomial completely.

Answer 1

i. No, the student’s solution is not correct.

ii. Let $$f(x) = 2x^{3} + 7x^{2} + 2x - 3$$.

By the Remainder Theorem, the remainder when $$f(x)$$ is divided by $$2x + 1$$ is

$$f\left(-\frac{1}{2}\right) = 2\left(-\frac{1}{2}\right)^3 + 7\left(-\frac{1}{2}\right)^2 + 2\left(-\frac{1}{2}\right) - 3$$

$$=-\frac{5}{2}\ne 0$$

Since the remainder is not zero, $$2x + 1$$ is not a factor.

Also,

$$f(-1) = -2 + 7 - 2 - 3 = 0,$$

so $$x + 1$$ is a factor and

$$f(-3) = -54 + 63 - 6 - 3 = 0,$$

so $$x + 3$$ is a factor. 

Hence, the complete factorisation is $$2x^{3} + 7x^{2} + 2x - 3 = (x + 1)(x + 3)(2x - 1).$$