Question

Statement1: The polynomial $P\left(x\right)=4x^3–3x^2+5x–6$ when divided by $x–1$ gives zero as the remainder.

Statement2: $(x–1)$ is a factor of the polynomial $P\left(x\right)=4x^3–3x^2+5x–6$.

• A. Both the statements are true; and statement 2 is the correct explanation of statement 1.
• B. Both the statements are true; and statement 2 is not the correct explanation of statement 1.
• C. Statement 1 is true and statement 2 is false.
• D. Statement 1 is false and statement 2 is true.

Answer 1

The correct option is A Both the statements are true; and statement 2 is the correct explanation of statement 1.

According to factor theorem when a polynomial $P\left(x\right)$ gives zero as the remainder when divided by (x – a), where ‘a’ is any real number, then (x – a) is the factor of the polynomial $P\left(x\right)$.

For the given polynomial $P\left(x\right)=4x^3–3x^2+5x–6$
$P\left(1\right)=4{\left(1\right)}^3–3{\left(1\right)}^2+5\left(1\right)–6=0$

Since $P\left(1\right)=0$,
(x-1) is a factor of $P\left(x\right)$.

When $P\left(x\right)$ is divided by (x-1), the remainder will be zero.

Thus, statement 2 is the correct explanation to statement 1.