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Factor Theorem

px = x3 + 8x2...

Question

$p (x) = x^{3} + 8 x^{2} - 7 x + 12$ and $g (x) = x - 1$ . If $p (x)$ is divided by $g (x)$ , it gives $q (x)$ and $r (x)$ as quotient and remainder respectively. If $a$ is the degree of $q (x)$ and $b$ is the degree of $r (x)$ , $(a - b) = ?$ .

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Solution

Remainder $= r (x) = p (1) = 1^{3} + 8 \times 1^{2} - 7 \times 1 + 12 = 14$
The degree of a polynomial is the highest degree of its terms when the polynomial is expressed in its canonical form consisting of a linear combination of monomials.
So, Degree of $r (x) = b = 0$
$q (x) = \frac{p (x)}{g (x)}$
Now , $p (x) = x^{3} + 8 x^{2} - 7 x + 12$
$g (x) = (x - 1)$
Performing long division

$x^{2} + 9 x + 2 x - 1) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ x^{3} + 8 x^{2} - 7 x + 12 x^{3} - x^{2} (-) (+) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 9 x^{2} - 7 x 9 x^{2} - 9 x (-) (+) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ + 2 x + 12 2 x - 2 (-) (+) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 14$

$∴ q (x) = \frac{p (x)}{g (x)} = x^{2} + 9 x + 2$
So, degree of $q (x) = a = 2$
Hence, $a - b = 2 - 0 = 2$

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Similar questions

$p (x) = x^{3} + 8 x^{2} - 7 x + 12$ and $g (x) = x - 1$ . If $p (x)$ is divided by $g (x)$ , it gives $q (x)$ and $r (x)$ as quotient and remainder respectively. If $a$ is the degree of $q (x)$ and $b$ is the degree of $r (x)$ , then, $(a - b) = ?$ .

Q. Who is correct?
Aakanksha:

q (x) = \frac{p (x) - r (x)}{g (x)}

Arjun:

q (x) = \frac{p (x)}{g (x)} - r (x)

Aman:

q (x) = p (x) - \frac{r (x)}{g (x)}

Anasuya:

q (x) = p (x) + \frac{r (x)}{g (x)}

Given,

p (x)

is the dividend,

g (x)

is the divisor,

q (x)

is the quotient, and

r (x)

is the remainder

Q. Find the quotient

q (x)

and remainder

r (x)

of the following when

f (x)

is divided by

g (x)

p (x) = x^{6} + x^{4} - x^{2} - 1

;

g (x) = x^{3} - x^{2} + x - 1

Q. Find the value of t, given that

p (x) = x^{3} - t x^{2} + 5 x - 3,

g (x) = x^{2} - 2,

q (x) = x - 3,

r (x) = 7 x - 9

Where,

p (x)

is the dividend,

g (x)

is the divisor,

q (x)

is the quotient, and

r (x)

is the remainder

If p(x) = g(x) × q(x) + r(x), p(x) and g(x) are any two polynomials with g(x) $\neq$ 0, then we can find polynomials q(x) and r(x) by division algorithm.We should stop the division process when

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p(x)=x3+8x2−7x+12 and g(x)=x−1. If p(x) is divided by g(x), it gives q(x) and r(x) as quotient and remainder respectively. If a is the degree of q(x) and b is the degree of r(x), (a−b)=?.

$p (x) = x^{3} + 8 x^{2} - 7 x + 12$ and $g (x) = x - 1$ . If $p (x)$ is divided by $g (x)$ , it gives $q (x)$ and $r (x)$ as quotient and remainder respectively. If $a$ is the degree of $q (x)$ and $b$ is the degree of $r (x)$ , $(a - b) = ?$ .