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Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following :
(i)
(ii)
(iii)
(i) quotient= x-3 , remainder= 7x-9
(ii) quotient= x2+x-3 , remainder= 8
(iii) quotient= -x2-2 , remainder= -5x+10
Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:
(i)
(ii)
(iii)
The polynomial 2t4 + 3t3 - 2t2 - 9t - 12 can be divided by the polynomial t2 - 3 = t2 + 0.t - 3 as follows:
Since the remainder is 0, t² - 3 is a factor of 2t4 + 3t3 - 2t2 - 9t - 12 .
(ii) The polynomial 3x4 + 5x3 - 7x2 + 2x + 2 can be divided by the polynomial x2 + 3x + 1 as follows:
Since the remainder is 0, x² + 3x + 1 is a factor of 3x4 + 5x3 - 7x2 + 2x + 2
(iii) The polynomial x5 - 4x3 + x2 + 3x + 1 can be divided by the polynomial x3 - 3x + 1 as follows:
Since the remainder is not equal to 0, x3 - 3x + 1 is not a factor of x5 - 4x3 + x2 + 3x + 1.
Obtain all other zeroes of , if two of its zeros are
and
.
The two zeroes given are
(x - √5/3) (x + √5/3) = x² - 5/3
When we divide 3x4 + 6x³ - 2x² - 10x -5 by x² - 5/3 we get 3x² + 6x + 3
Now, 3x² + 6x + 3 = 0.
∴ 3x² + 3x + 3x +3 = 0
∴ 3x (x + 1) + 3 (x +1) = 0
∴ (3x + 3) (x + 1) = 0
∴ 3 (x + 1) (x + 1) = 0
∴ x + 1 = 0
∴ x = -1
The other two zeroes are -1 and -1
On dividing by a polynomial g(x), the quotient and remainder were
and
, repectively. Find g(x)
Here, Dividend = x³ - 3x² + x + 2, Quotient = ( x - 2 ) & Remainder
= ( - 2x + 4)
∴ Quotient × Divisor + Remainder
= Dividend.
⇒ ( x - 2 ) × g(x) + ( - 2x + 4 )
= x³ - 3x² + x + 2
⇒ ( x - 2 ) × g(x) = x³ - 3x² + x + 2 + 2x - 4
⇒ g(x) = x³ - 3x² + 3x - 2/x - 2
∴ g(x) = x² - x + 1
Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and
(i) (ii)
(iii)
(i) Degree of quotient will be equal to degree of dividend when divisor is constant.
Let us consider the division of 18x2 + 3x + 9 by 3.
Here, p(x) = 18x2 + 3x + 9 and g(x) = 3
q(x) = 6x2 + x + 3 and r(x) = 0
Here, degree of p(x) and q(x) is the same which is 2.
Checking:
p(x) = g(x) x q(x) + r(x)
Thus, the division algorithm is satisfied.
(ii) Let us consider the division of 2x4 + 2x by 2x3,
Here, p(x) = 2x4 + 2x and g(x) = 2x3
q(x) = x and r(x) = 2x
Clearly, the degree of q(x) and r(x) is the same which is 1.
Checking,
p(x) = g(x) x q(x) + r(x)
2x4 + 2x = (2x3 ) x x + 2x
2x4 + 2x = 2x4 + 2x
Thus, the division algorithm is satisfied.
(iii) Degree of remainder will be 0 when remainder obtained on division is a constant.
Let us consider the division of 10x3 + 3 by 5x2.
Here, p(x) = 10x3 + 3 and g(x) = 5x2
q(x) = 2x and r(x) = 3
Clearly, the degree of r(x) is 0.
Checking:
p(x) = g(x) x q(x) + r(x)
10x3 + 3 = (5x2 ) x 2x + 3
10x3 + 3 = 10x3 + 3
Thus, the division algorithm is satisfied.
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