In each of the following, using the remainder theorem, find the remain

1.	In each of the following, using the remainder theorem, find the remainder when f(x) is divided by g(x):f(x) = 4x3 - 12x2 +14x - 3, g(x) = 2x - 1
Answer» We have, f(x) = 4x³ - 12x² + 14x - 3 and g(x) = 2x - 1 Therefore, by remainder theorem when f (x) is divided by g (x) = 2 (x - \(\frac{1}{2})\), the remainder is equal to f \((\frac{1}{2})\) Now, f(x) = 4x³ -12x² + 14x - 3 f\((\frac{1}{2})\) = 4 \((\frac{1}{2})^{3}-12(\frac{1}{2})^{2}+14(\frac{1}{2})-3\) = (4 * \(\frac{1}{8})\) - (12 * \(\frac{1}{4})+7-3\) = \(\frac{1}{2}-3+7-3\) = \(\frac{3}{2}\) Hence, required remainder is \(\frac{3}{2}\)

In each of the following, using the remainder theorem, find the remainder when f(x) is divided by g(x):f(x) = 4x3 - 12x2 +14x - 3, g(x) = 2x - 1

Answer»

We have,

f(x) = 4x³ - 12x² + 14x - 3 and g(x) = 2x - 1

Therefore, by remainder theorem when f (x) is divided by g (x) = 2 (x - \(\frac{1}{2})\), the remainder is equal to f \((\frac{1}{2})\)

Now, f(x) = 4x³ -12x² + 14x - 3

f\((\frac{1}{2})\) = 4 \((\frac{1}{2})^{3}-12(\frac{1}{2})^{2}+14(\frac{1}{2})-3\)

= (4 * \(\frac{1}{8})\) - (12 * \(\frac{1}{4})+7-3\)

= \(\frac{1}{2}-3+7-3\)

= \(\frac{3}{2}\)

Hence, required remainder is \(\frac{3}{2}\)