Divide the following polynomials using long division.
Problem 1 :
(2x3 - 6x2 + 5x + 4) ÷ (x - 2)
Problem 2 :
(4x3 - 5x2 + 6x - 2) ÷ (x - 1)
Problem 3 :
(x3 - 7x2 - x + 6) ÷ (x + 2)
Problem 4 :
(10- 4x + 3x2) ÷ (x - 2)
Problem 5 :
f(x) = -2(x2 + 7x - 3) - a(x + 2) + 1
In the polynomial f(x) is defined above, a is constant. IF f(x) is divisible by x, what is the value of a ?
a) -5/2 b) -3 c) 7/2 d) 5
Problem 6 :
If (-2x2 + 5x + 10)/(-x + 4) = (2x + 3) - A/(-x + 4), what is the value of A ?
Problem 7 :
(x3 - x2 + 3x - 3) ÷ (x - 1)
Which of the follwoing is the quotient of the expression above ?
a) x2 - 3 b) x2 + 3 c) x2 - 2x d) x2 - 2x + 3
Problem 8 :
The function f is defined by a polynomial some values of x and f(x) are shown in the table above. What is the remainder when f(x) is divided by x + 2 ?
a) -10.5 b) -8 c) 2.5 d) 7.5
Problem 9 :
g(x) = -(x2 - 6x + 5) - 4(x - c)
In the polynomial g(x) defined above, c is a constant. If g(x) is divisible by x + 1, what is the value of c ?
a) 1 b) 2 c) 3 d) 4
Problem 10 :
Which of the following is not a factor of the polynomial p(x) = 2x3 - 5x2 - 4x + 3 ?
a) 2x - 1 b) x + 1 c) x - 3 d) x - 1
1. Answer :
(2x3 - 6x2 + 5x + 4) ÷ (x - 2)
Let P(x) = 2x3 - 6x2 + 5x + 4 and g(x) = x - 2
To divide the given polynomial by x - 2, we have divide the first term of the polynomial P(x) by the first term of the polynomial g(x).
If we divide 2x3 by x, we get 2x2. Now we have to multiply this 2x2 by x - 2. From this we get 2x3 - 4x2.
Now we have to subtract 2x3 - 4x2 from the given polynomial. So we get -2x2 + 5x + 4.
Now we have to subtract 2x3 - 4x2 from the given polynomial. So we get -2x2 + 5x + 4.
repeat this process until we get the degree of p(x) ≥ degree of g(x).
So,
Quotient = 2x2 - 2x + 1
Remainder = 6
2. Answer :
(4x3 - 5x2 + 6x - 2) ÷ (x - 1)
So,
Quotient = 4x2 - x + 5
Remainder = 3
3. Answer :
(x3 - 7x2 - x + 6) ÷ (x + 2)
So,
Quotient = x2 - 9x + 17
Remainder = -28
4. Answer :
(10- 4x + 3x2) ÷ (x - 2)
Let us first write the terms of each polynomial in descending order ( or ascending order).
Thus, the given problem becomes (10- 4x + 3x2) ÷ (x - 2)
f(x) = 10- 4x + 3x2
= 3x2 - 4x + 10
g(x) = x - 2
Step 1 :
In the first step, we are going to divide the first term of the dividend by the first first term of the divisor.
After changing the signs, +3x2 and -3x2 will get canceled. By simplifying, we get 2x + 10.
Step 2 :
In the second step again we are going to divide the first term that is 2x by the first term of divisor that is x.
So,
Quotient = 3x + 2
Remainder = 14
5. Answer :
f(x) = -2(x2 + 7x - 3) - a(x + 2) + 1
Since the polynomial is divisible by x, that is x - 0, we will get the remainder as 0.
We may solve this problem using remainder theorem.
f(0) = -2(02 + 7(0) - 3) - a(0 + 2) + 1
0 = -2(-3) - 2a + 1
0 = 6 - 2a + 1
0 = 7 - 2a
2a = 7
a = 7/2
So, the value of a is 7/2, otpion c is correct.
6. Answer :
If (-2x2 + 5x + 10)/(-x + 4) = (2x + 3) - A/(-x + 4), what is the value of A ?
(2x + 3) - [2/(-x + 4)]
Comparing this with (2x + 3) - A/(-x + 4), the value of A is 2.
7. Answer :
(x3 - x2 + 3x - 3) ÷ (x - 1)
Dividing the polynomial by the linear,
The required quotient is x2 + 3.
8. Answer :
By observing the table, applying the value of x as -4, we get the remainder -10.5
Dividing the polynomial by x + 4, we get the remainder as -10.5
Dividing the polynomial by x + 2, we get the remainder as 2.5
So, option c is correct.
9. Answer :
g(x) = -(x2 - 6x + 5) - 4(x - c)
Dividing the polynomial by x + 1, that is x + 1 = 0
x = -1
Applying -1 as x, we get
g(x) = -(x2 - 6x + 5) - 4(x - c)
Since it is divisible by -1, we ge the remainder as 0
g(-1) = -((-1)2 - 6(-1) + 5) - 4(-1 - c)
0 = -(1 + 6 + 5) + 4 + 4c
0 = -12 + 4 + 4c
0 = -8 + 4c
4c = 8
c = 8/4
c = 2
So, the value of c is 2, option b is correct.
10. Answer :
p(x) = 2x3 - 5x2 - 4x + 3
Dividing the polynomial by 2x + 1, 2x + 1 = 0
2x = -1
x = -1/2
Dividing the polynomial by x + 1, we get the remainder 0. So, option b is correct.
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