Question

Find the remainder when x³-4x²+12x+7 is divided by x+½

Answer 1

here is your answer how is that both are correct..first one is direct division method and second one remainder theorem method

Answer 2

The remainder when x³ - 4x² + 12x + 7 is divided by x+½, is -1/8.

Given,

Two polynomial expressions:

x³ - 4x² + 12x + 7,

x+½.

To find,

Remainder when x³ - 4x² + 12x + 7 is divided by x+½.

Solution,

According to the remainder theorem, when a polynomial, say p(x) is divided by the other linear type of polynomial, say q(x), and q(x) has x = a, as a zero, then the remainder is given by

r = p(a)

Here, we can see that

x³ - 4x² + 12x + 7 is divided by x+½.

So we can say,

p(x) = x³ - 4x² + 12x + 7, and

q(x) = x+½.

We can see that putting

x+½ = 0

⇒ x = -½.

So, the zero of x+½ is

$x=-\frac{1}{2}.$

Thus, the remainder for the given division will be given by p(-½).

$p(x=-\frac{1}{2}) = p(-\frac{1}{2})= (-\frac{1}{2})^3 - 4(-\frac{1}{2})^{2} + 12(-\frac{1}{2}) + 7$

$\implies p(-\frac{1}{2})= (-\frac{1}{8}) - 4(\frac{1}{4}) - 12(\frac{1}{2}) + 7$

$\implies p(-\frac{1}{2})= -\frac{1}{8} - 1 - 6 + 7$

$\implies p(-\frac{1}{2})= -\frac{1}{8}$

$\implies remainder =-\frac{1}{8}.$

Therefore, the remainder when x³ - 4x² + 12x + 7 is divided by x+½, is -1/8.

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