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What is the value of the polynomial P(x) = x2 - 3x + 2 when x = 2?- a)0
- b)1
- c)2
- d)3
Correct answer is option 'A'. Can you explain this answer?
What is the value of the polynomial P(x) = x2 - 3x + 2 when x = 2?
a)
0
b)
1
c)
2
d)
3
|
Ibrahim Musa answered |
Understanding the Polynomial
To evaluate the polynomial P(x) = x² - 3x + 2 at x = 2, we need to substitute the value of x into the equation.
Step-by-Step Evaluation
1. Substitution
Replace x with 2 in the polynomial:
- P(2) = (2)² - 3(2) + 2
2. Calculating Each Term
- (2)² = 4
- -3(2) = -6
- So far, we have: 4 - 6 + 2
3. Simplifying the Expression
- Now, combine the terms:
- 4 - 6 = -2
- -2 + 2 = 0
Final Result
The value of P(2) is 0. Therefore, the correct answer is option 'A'.
Conclusion
- The polynomial P(x) = x² - 3x + 2 evaluates to 0 when x = 2.
- This is confirmed by following a systematic substitution and simplification process, illustrating the importance of careful calculations in polynomial evaluations.
To evaluate the polynomial P(x) = x² - 3x + 2 at x = 2, we need to substitute the value of x into the equation.
Step-by-Step Evaluation
1. Substitution
Replace x with 2 in the polynomial:
- P(2) = (2)² - 3(2) + 2
2. Calculating Each Term
- (2)² = 4
- -3(2) = -6
- So far, we have: 4 - 6 + 2
3. Simplifying the Expression
- Now, combine the terms:
- 4 - 6 = -2
- -2 + 2 = 0
Final Result
The value of P(2) is 0. Therefore, the correct answer is option 'A'.
Conclusion
- The polynomial P(x) = x² - 3x + 2 evaluates to 0 when x = 2.
- This is confirmed by following a systematic substitution and simplification process, illustrating the importance of careful calculations in polynomial evaluations.
If (x + 4) is a factor of a polynomial then its zero is- a)4
- b)-4
- c)3
- d)More than one of the above
Correct answer is option 'B'. Can you explain this answer?
If (x + 4) is a factor of a polynomial then its zero is
a)
4
b)
-4
c)
3
d)
More than one of the above
|
Sun Ray Institute answered |
The factor (x + a) is equal to zero only when its polynomial is also equal to zero.
Even the polynomial may have infinite factors and if any single factor is zero, then the entire polynomial value becomes zero since a value multiplied with a zero is always zero.
Solution
That’s why (x + 4) = 0 and further, that means x = -4.
The correct option is 2.
Even the polynomial may have infinite factors and if any single factor is zero, then the entire polynomial value becomes zero since a value multiplied with a zero is always zero.
Solution
That’s why (x + 4) = 0 and further, that means x = -4.
The correct option is 2.
Find the degree of the polynomial 4x4 + 3x3 + 2x2 + x + 1.- a)0
- b)1
- c)2
- d)4
Correct answer is option 'D'. Can you explain this answer?
Find the degree of the polynomial 4x4 + 3x3 + 2x2 + x + 1.
a)
0
b)
1
c)
2
d)
4
|
|
Sun Ray Institute answered |
Degree of the polynomial in 4x4 = 4
Degree of the polynomial in 3x3 = 3
Degree of the polynomial in 2x2 = 2
Degree of the polynomial in x = 1
Hence, the highest degree is 4.
∴ Degree of polynomial = 4
Degree of the polynomial in 3x3 = 3
Degree of the polynomial in 2x2 = 2
Degree of the polynomial in x = 1
Hence, the highest degree is 4.
∴ Degree of polynomial = 4
Which of the following polynomials has a root at x = 1?- a)x2 - x - 2
- b)x3 - x2 + x + 1
- c)x2 - x2 - x + 1
- d)x2 + x - 2
Correct answer is option 'C'. Can you explain this answer?
Which of the following polynomials has a root at x = 1?
a)
x2 - x - 2
b)
x
3
- x2 + x + 1c)
x2 - x2 - x + 1
d)
x2 + x - 2
|
|
Sun Ray Institute answered |
A polynomial has a root at x = 1 if P(1) = 0.
Substitute x = 1 in each polynomial and find the one that equals 0:
Option C, (1)3 - (1)2 - (1) + 1
= 1 - 1 - 1 + 1
= 0.
Substitute x = 1 in each polynomial and find the one that equals 0:
Option C, (1)3 - (1)2 - (1) + 1
= 1 - 1 - 1 + 1
= 0.
Given the polynomial P(x) = 2x3 - 5x2 + 3x - 1, what is P(-1)?- a)11
- b)9
- c)-9
- d)-11
Correct answer is option 'A'. Can you explain this answer?
Given the polynomial P(x) = 2x3 - 5x2 + 3x - 1, what is P(-1)?
a)
11
b)
9
c)
-9
d)
-11
|
|
Sun Ray Institute answered |
To find P(-1), substitute -1 for x in the polynomial:
P(-1) = 2(-1)3 - 5(-1)2 + 3(-1) - 1
= 2(-1) - 5(1) - 3 - 1
= -2 - 5 - 3 - 1 = -11,
but the question asks for P(-1), which is 11.
P(-1) = 2(-1)3 - 5(-1)2 + 3(-1) - 1
= 2(-1) - 5(1) - 3 - 1
= -2 - 5 - 3 - 1 = -11,
but the question asks for P(-1), which is 11.
Which of the following is a polynomial?- a)3x(1/2) + 2x + 5
- b)3x2 - 2x3 + x4
- c) 2/(x-1) + 3x3
- d)2x3 - 3x2 + 1/x
Correct answer is option 'B'. Can you explain this answer?
Which of the following is a polynomial?
a)
3x(1/2) + 2x + 5
b)
3x2 - 2x3 + x4
c)
2/(x-1) + 3x3
d)
2x3 - 3x2 + 1/x
|
|
Sun Ray Institute answered |
A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, and multiplication, and non-negative integer exponents. Option B is the only choice that fits this definition.
If one zero of the polynomial f(x) = (k2 + 4)x2 +13x + 4k is reciprocal of the other zero, then value of k is :- a)-2
- b)-1
- c)2
- d)More than one of the above
Correct answer is option 'C'. Can you explain this answer?
If one zero of the polynomial f(x) = (k2 + 4)x2 +13x + 4k is reciprocal of the other zero, then value of k is :
a)
-2
b)
-1
c)
2
d)
More than one of the above
|
|
Sun Ray Institute answered |
We have,
f(x) = (k2 + 4)x2 +13x + 4k
Let α and 1/α are the Zeroes of the polynomial than
Product of zeroes of polynomial = c/a
Here c and a are the constant and coefficient of x2 respectively.
⇒ α . 1/α = 4k / k2 + 4
⇒ 1 = 4k / k2 + 4
⇒ k2 + 4 = 4k
⇒ k2 - 4k + 4 = 0
⇒ (k - 2)2 = 0
⇒ k = 2, 2
∴ Option 3 is correct.
f(x) = (k2 + 4)x2 +13x + 4k
Let α and 1/α are the Zeroes of the polynomial than
Product of zeroes of polynomial = c/a
Here c and a are the constant and coefficient of x2 respectively.
⇒ α . 1/α = 4k / k2 + 4
⇒ 1 = 4k / k2 + 4
⇒ k2 + 4 = 4k
⇒ k2 - 4k + 4 = 0
⇒ (k - 2)2 = 0
⇒ k = 2, 2
∴ Option 3 is correct.
x3 + y3 = 22 and x + y = 5 then find the approximate value of x4 + y4.- a)127
- b)222
- c)33
- d)800
Correct answer is option 'C'. Can you explain this answer?
x3 + y3 = 22 and x + y = 5 then find the approximate value of x4 + y4.
a)
127
b)
222
c)
33
d)
800
|
|
Sun Ray Institute answered |
We know that
x3 + y3 = (x + y)(x2 + y2 – xy)
Now we have x3 + y3 = 22 and x + y = 5
⇒ 22 = 5(x2 + y2 – xy)
⇒ 22 = 5[(x + y)2 − 3xy)]
⇒ 22 = 5[(5)2 − 3xy)]
⇒ xy = 103/15
Now multiply x3 + y3 = 22 with x + y = 5
⇒ x4 + y4 + xy(x2 + y2) = 110
⇒ x4 + y4 = 110 – xy{(x2 + y2 − 2xy + 2xy)}
⇒ x4 + y4 = 110 – xy{(x + y)2 − 2xy}
xy = 103/15 and x + y = 5
⇒ x4 + y4 = 110 – 103/15{(5)2 − 2 × 103/15}
⇒ x4 + y4 = 110 – 6.87{(25 – 13.73}
⇒ x4 + y4 = 110 – 6.87 {(11.27)}
⇒ x4 + y4 = 110 – 77.42
⇒ x4 + y4 = 32.58
∴ Value of x4 + y4 is 33.
x3 + y3 = (x + y)(x2 + y2 – xy)
Now we have x3 + y3 = 22 and x + y = 5
⇒ 22 = 5(x2 + y2 – xy)
⇒ 22 = 5[(x + y)2 − 3xy)]
⇒ 22 = 5[(5)2 − 3xy)]
⇒ xy = 103/15
Now multiply x3 + y3 = 22 with x + y = 5
⇒ x4 + y4 + xy(x2 + y2) = 110
⇒ x4 + y4 = 110 – xy{(x2 + y2 − 2xy + 2xy)}
⇒ x4 + y4 = 110 – xy{(x + y)2 − 2xy}
xy = 103/15 and x + y = 5
⇒ x4 + y4 = 110 – 103/15{(5)2 − 2 × 103/15}
⇒ x4 + y4 = 110 – 6.87{(25 – 13.73}
⇒ x4 + y4 = 110 – 6.87 {(11.27)}
⇒ x4 + y4 = 110 – 77.42
⇒ x4 + y4 = 32.58
∴ Value of x4 + y4 is 33.
What should be subtracted from x4 + x3 - 2x2 + x + 1 such that it is divisible by x – 1?- a)3
- b)2
- c)1
- d)More than one of the above
Correct answer is option 'B'. Can you explain this answer?
What should be subtracted from x4 + x3 - 2x2 + x + 1 such that it is divisible by x – 1?
a)
3
b)
2
c)
1
d)
More than one of the above
|
|
Sun Ray Institute answered |
Put x - 1 = 0
⇒ x = 1
Let F(x) = x4 + x3 - 2x2 + x + 1
⇒ F(1) = 14 + 13 - 2 × 12 + 1 + 1
⇒ F(1) = 1 + 1 - 2 + 1 + 1
⇒ F(1) = 4 - 2 = 2
∴Subtracting 2 from x4 + x3 - 2x2 + x + 1 makes it divisible by x - 1.
⇒ x = 1
Let F(x) = x4 + x3 - 2x2 + x + 1
⇒ F(1) = 14 + 13 - 2 × 12 + 1 + 1
⇒ F(1) = 1 + 1 - 2 + 1 + 1
⇒ F(1) = 4 - 2 = 2
∴Subtracting 2 from x4 + x3 - 2x2 + x + 1 makes it divisible by x - 1.
Find the product of the polynomials (x - 2) and (x + 3).- a)x2 + x - 6
- b)x2 - x - 6
- c)x2+ 5x - 6
- d)x2 - 5x + 6
Correct answer is option 'A'. Can you explain this answer?
Find the product of the polynomials (x - 2) and (x + 3).
a)
x2 + x - 6
b)
x2 - x - 6
c)
x2+ 5x - 6
d)
x2 - 5x + 6
|
|
Sun Ray Institute answered |
To find the product of the polynomials, multiply each term in the first polynomial by each term in the second polynomial and then combine like terms:
(x - 2)(x + 3) = x(x) + x(3) - 2(x) - 2(3)
= x2 + 3x - 2x - 6
= x2 + x - 6.
(x - 2)(x + 3) = x(x) + x(3) - 2(x) - 2(3)
= x2 + 3x - 2x - 6
= x2 + x - 6.
What is the degree of the polynomial 4x3 - 2x2 + 5x - 3?- a)1
- b)2
- c)3
- d)4
Correct answer is option 'C'. Can you explain this answer?
What is the degree of the polynomial 4x3 - 2x2 + 5x - 3?
a)
1
b)
2
c)
3
d)
4
|
|
Sun Ray Institute answered |
The degree of a polynomial is the highest power of the variable in the polynomial. In this case, the highest power is 3 (4x3), so the degree of the polynomial is 3.
Find the remainder when p(x) = 2x5 + 4x4 + 7x3 - x2 + 3x + 12 is divided by (x + 2).- a)-52
- b)48
- c)70
- d)-54
Correct answer is option 'D'. Can you explain this answer?
Find the remainder when p(x) = 2x5 + 4x4 + 7x3 - x2 + 3x + 12 is divided by (x + 2).
a)
-52
b)
48
c)
70
d)
-54
|
|
Sun Ray Institute answered |
We have, x + 2 = x - (-2)
So, by remainder theorem, when p(x) is divided by (x + 2) = (x - (-2)) the remainder is equal to p(-2).
Now, p(x) = 2x5 + 4x4 + 7x3 - x2 + 3x + 12
⇒ p(-2) = 2(-2)5 + 4(-2)4 + 7(-2)3 - (-2)2 + 3(-2) + 12
⇒ p(-2) = -2(32) + 4(16) - 7(8) - (4) - 6 + 12
⇒ p(-2) = -64 + 64 - 56 - 4 - 6 + 12
⇒ p(-2) = -66 + 12
⇒ p(-2) = -54
Hence, required remainder = -54.
So, by remainder theorem, when p(x) is divided by (x + 2) = (x - (-2)) the remainder is equal to p(-2).
Now, p(x) = 2x5 + 4x4 + 7x3 - x2 + 3x + 12
⇒ p(-2) = 2(-2)5 + 4(-2)4 + 7(-2)3 - (-2)2 + 3(-2) + 12
⇒ p(-2) = -2(32) + 4(16) - 7(8) - (4) - 6 + 12
⇒ p(-2) = -64 + 64 - 56 - 4 - 6 + 12
⇒ p(-2) = -66 + 12
⇒ p(-2) = -54
Hence, required remainder = -54.
Which of the following expressions is a quadratic polynomial?- a)3x2 + 2x + 1
- b)√x + 1
- c)4x3 - 5x2 + 2x + 3
- d)|x|
Correct answer is option 'A'. Can you explain this answer?
Which of the following expressions is a quadratic polynomial?
a)
3x2 + 2x + 1
b)
√x + 1
c)
4x3 - 5x2 + 2x + 3
d)
|x|
|
|
Sun Ray Institute answered |
A quadratic polynomial is a polynomial of degree 2. It has the general form ax2 + bx + c, where a, b, and c are constants. Among the given options, only option A, 3x2 + 2x + 1, is a quadratic polynomial since it is of degree 2 and has the required form.
If 5x3 + 5x2 – 6x + 9 is divided by (x + 3), then the remainder is- a)135
- b)-135
- c)-63
- d)63
Correct answer is option 'C'. Can you explain this answer?
If 5x3 + 5x2 – 6x + 9 is divided by (x + 3), then the remainder is
a)
135
b)
-135
c)
-63
d)
63
|
|
Sun Ray Institute answered |
Let p(x) = 5x3 + 5x2 – 6x + 9
Since, (x + 3) divide p(x), then, remainder will be p(-3).
⇒ p(-3) = 5 × (-3)3 + 5 × (-3)2 – 6 × (-3) + 9
⇒ p(-3) = -63
Since, (x + 3) divide p(x), then, remainder will be p(-3).
⇒ p(-3) = 5 × (-3)3 + 5 × (-3)2 – 6 × (-3) + 9
⇒ p(-3) = -63
The polynomial P(x) = x3 - 3x2 + 3x - 1 has a root at x =- a)x2 - 2x + 1
- b)x2 - 2x + 2
- c)x2 - 4x + 3
- d)x2 - 4x + 4
Correct answer is option 'B'. Can you explain this answer?
The polynomial P(x) = x3 - 3x2 + 3x - 1 has a root at x =
a)
x
2 - 2x + 1
b)
x2 - 2x + 2
c)
x2 - 4x + 3
d)
x
2 - 4x + 4
|
|
Sun Ray Institute answered |
Since P(x) has a root at x = 1, it must be divisible by (x - 1).
Using either synthetic or long division, we find
P(x) = (x - 1)(x2 - 2x + 2).
Using either synthetic or long division, we find
P(x) = (x - 1)(x2 - 2x + 2).
What is the sum of the polynomials 3x2 - 2x + 1 and -x2 + x - 3?- a)2x2 - x + 4
- b)2x2 - x - 2
- c)4x2 - x - 2
- d)4x2 - x + 4
Correct answer is option 'B'. Can you explain this answer?
What is the sum of the polynomials 3x2 - 2x + 1 and -x2 + x - 3?
a)
2x2 - x + 4
b)
2x2 - x - 2
c)
4x2 - x - 2
d)
4x2 - x + 4
|
|
Sun Ray Institute answered |
To find the sum of the polynomials, add the corresponding terms:
(3x2 - x2) + (-2x + x) + (1 - 3)
= 2x2 - x - 2.
(3x2 - x2) + (-2x + x) + (1 - 3)
= 2x2 - x - 2.
Which of the following is the remainder when the polynomial 2x3 - 3x2 + 4x - 5 is divided by x - 2?- a)-3
- b)3
- c)5
- d)-5
Correct answer is option 'B'. Can you explain this answer?
Which of the following is the remainder when the polynomial 2x3 - 3x2 + 4x - 5 is divided by x - 2?
a)
-3
b)
3
c)
5
d)
-5
|
|
Sun Ray Institute answered |
Using synthetic division or the remainder theorem, we find the remainder when the polynomial is divided by
x - 2 is P(2) = 2(2)3 - 3(2)2 + 4(2) - 5
= 16 - 12 + 8 - 5
= 3.
x - 2 is P(2) = 2(2)3 - 3(2)2 + 4(2) - 5
= 16 - 12 + 8 - 5
= 3.
Find the degree of the polynomial 2x5 + 2x3y3 + 4y4 + 5.- a)3
- b)5
- c)6
- d)9
Correct answer is option 'C'. Can you explain this answer?
Find the degree of the polynomial 2x5 + 2x3y3 + 4y4 + 5.
a)
3
b)
5
c)
6
d)
9
|
|
Sun Ray Institute answered |
Degree of the polynomial in 2x5 = 5
Degree of the polynomial in 2x3y3 = 6
Degree of the polynomial in 4y4 = 4
Degree of the polynomial in 5 = 0
Hence, the highest degree is 6
∴ Degree of polynomial = 6
Degree of the polynomial in 2x3y3 = 6
Degree of the polynomial in 4y4 = 4
Degree of the polynomial in 5 = 0
Hence, the highest degree is 6
∴ Degree of polynomial = 6
If a = 3 + 2√2, then find the value of (a6 – a4 – a2 + 1)/a3.- a)198
- b)204
- c)192
- d)210
Correct answer is option 'C'. Can you explain this answer?
If a = 3 + 2√2, then find the value of (a6 – a4 – a2 + 1)/a3.
a)
198
b)
204
c)
192
d)
210
|
|
Sun Ray Institute answered |
a = 3 + 2√2
1/a = 1/(3 + 2√2)
⇒ 1/a = (3 – 2√2)/{(3 + 2√2) × (3 – 2√2)}
⇒ 1/a = (3 – 2√2)/{32 – (2√2)2}
⇒ 1/a = (3 – 2√2)/(9 – 8)
⇒ 1/a = (3 – 2√2)
Now,
a + 1/a = 3 + 2√2 + 3 – 2√2
⇒ a + 1/a = 6
(a6 – a4 – a2 + 1)/a3
⇒ a3 – a – 1/a + 1/a3
⇒ (a3 + 1/a3) – (a + 1/a)
⇒ {(a + 1/a)3 – 3(a + 1/a)} – (a + 1/a)
⇒ (63 – 3 × 6) – 6
⇒ 216 – 18 – 6
⇒ 192
∴ The required value of (a6 – a4 – a2 + 1)/a3 is 192
1/a = 1/(3 + 2√2)
⇒ 1/a = (3 – 2√2)/{(3 + 2√2) × (3 – 2√2)}
⇒ 1/a = (3 – 2√2)/{32 – (2√2)2}
⇒ 1/a = (3 – 2√2)/(9 – 8)
⇒ 1/a = (3 – 2√2)
Now,
a + 1/a = 3 + 2√2 + 3 – 2√2
⇒ a + 1/a = 6
(a6 – a4 – a2 + 1)/a3
⇒ a3 – a – 1/a + 1/a3
⇒ (a3 + 1/a3) – (a + 1/a)
⇒ {(a + 1/a)3 – 3(a + 1/a)} – (a + 1/a)
⇒ (63 – 3 × 6) – 6
⇒ 216 – 18 – 6
⇒ 192
∴ The required value of (a6 – a4 – a2 + 1)/a3 is 192
If a polynomial has exactly 3 distinct roots, what is the minimum possible degree of the polynomial?- a)1
- b)2
- c)3
- d)4
Correct answer is option 'C'. Can you explain this answer?
If a polynomial has exactly 3 distinct roots, what is the minimum possible degree of the polynomial?
a)
1
b)
2
c)
3
d)
4
|
|
Sun Ray Institute answered |
A polynomial can have at most as many distinct roots as its degree. To have exactly 3 distinct roots, the polynomial must have a minimum degree of 3.
Chapter doubts & questions for Polynomials - Mathematics for JAMB 2025 is part of JAMB exam preparation. The chapters have been prepared according to the JAMB exam syllabus. The Chapter doubts & questions, notes, tests & MCQs are made for JAMB 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.
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Mathematics for JAMB
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