1. Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainde
in each of the following:
() p(x)=x³-3x²+5x-3, g(x)=x²-2
(i) p()= x4 -3x+4r +5, 8(x)=x²+1-X
(ii) p(x)=x4 -5x +6, g(x) = 2-x²
Answers
Answer:
Let the given polynomial be p(x)=x
3
+13x
2
+31x−45.
We will now substitute various values of x until we get p(x)=0 as follows:
Forx=0
p(0)=(0)
3
+13(0)
2
+(31×0)−45=0+0+0−45=−45
=0
∴p(0)
=0
Forx=1
p(1)=(1)
3
+13(1)
2
+(31×1)−45=1+13+31−45=45−45=0
∴p(1)=0
Thus, (x−1) is a factor of p(x).
Now,
p(x)=(x−1)⋅g(x).....(1)
⇒g(x)=
(x−1)
p(x)
Therefore, g(x) is obtained by after dividing p(x) by (x−1) as shown in the above image:
From the division, we get the quotient g(x)=x
2
+14x+45 and now we factorize it as follows:
x
2
+14x+45
=x
2
+9x+5x+45
=x(x+9)+5(x+9)
=(x+5)(x+9)
From equation 1, we get p(x)=(x−1)(x+5)(x+9).
Hence, x
3
+13x
2
+31x−45=(x−1)(x+5)(x+9).
Answer:
Using division algorithm, find the quotient and remainder on dividing f(x) and g(x) where,
f(x) = 6x3 + 13x2 + x – 2 and g(x) = 2x + 1.
2. If the polynomial 6x4 + 8x3 + 17x2 +21x + 7 is divided by another polynomial 3x2 + 4x + 1,
the remainder comes out to be (ax + b), find a and b.
3. If the polynomial x4 – 6x3 + 16x2 – 25x + 10 is divided by (x2 – 2x + k) the remainder comes
out to be x + a, find k and a.
4. If 2 and -3 are the zeroes of the quadratic polynomial x2 + (a + 1) x + b; then find the values
of a.
5. It being given that 1 is one of the zeroes of the polynomial 7x – x3 – 6. Find its other zeroes.
6. If the zeroes of the polynomial x2 + px + q are double in value to the zeroes of 2x2 – 5x – 3,
find the value of p and q.
7. Divide x4 – 3x2 + 4x + 5 by x2 – x + 1, find quotient and remainder.
8. Find the zeroes of the following polynomials by factorisation method and verify the
relationship between the zeroes and coefficients of the polynomials.
a) 4x2 + 5 √2 x – 3 (b) 2x2 – (1 + 2√2)s + √2
c) v2 + 4√3 v – 15 (d) y2 +
3
2
√5 y – 5
e) 7y2 -
11
3
y -
2
3
9. Verify that 2, 1, 1 are zeroes of the polynomial x3 – 4x2 + 5x – 2. Also verify the relationship
between the zeroes and the coefficients.
10. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time
and product of its zeroes are 2, -7, -14 respectively.
11. α, β, γ are zeroes of cubic polynomial x3 – 12x2 + 44x + c. If 2β = α + γ, find the value of c.
12. Two zeroes of cubic polynomial ax3 + 3x2 – bx – 6 are -1 and -2. Find the third zero and the
value of a and b.
13. α, β, γ are zeroes of cubic polynomial x3 – 2x2 + qx – r. If α + β = 0 then show that 2q = r.
14. α, β, γ are zeroes of cubic polynomial x3 + px2 + qx + 2 such that α. β + 1 = 0. Find the value
of 2p + q + 5.
15. Write a quadratic polynomial, the sum and product of whose zeroes are 3 and -2.
16. If x + a is a factor of 2x2 + 2ax + 5x + 10, find a.
17. For what value of k, (-4) is a zero of the polynomial x2 – x – (2k + 2)?
18. If 1 is a zero of polynomial p(x) = ax2 – 3 (a – 1) – 1, then find the value of a.
19. If α, β are the zeroes of the polynomial 2y2 + 7y + 5, write the value of α + β + αβ.
20. If the polynomial 6x4 + 8x3 – 5x2 + ax + b is exactly divisible by the polynomial 2x2 – 5, then
find the values of a and b.
21. If the polynomial 6x4 + 8x3 + 17x2 + 21x + 7 is divided by another polynomial 3x2 + 4x + 1,
then what will be quotient and remainder?
22. Find the zeroes of the polynomial x4 – 7x2 + 12 if it is given that two of its zeroes are √3
and - √3.
23. Obtain all other zeroes of the polynomial x4 – 3x3 – x2 + 9x -6, if two of its zeroes are √3
and -
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