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M.M. 30
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Q1: Which of the following is NOT a polynomial? (1 Mark)
(a) 2x + 3y
(b) 4x^{2}  7x + 1
(c) √2x + 1
(d) x^{3}  5x^{2} + 4x  7
Ans: (c)
A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, where the exponents of variables are whole numbers (nonnegative integers). The option √2x + 1 contains a square root (√) in the expression, making it not a polynomial. All other options are polynomials.
Q2: If a polynomial has exactly three terms, what is it called? (1 Mark)
(a) Monomial
(b) Binomial
(c) Trinomial
(d) Polynomial
Ans: (c)
A polynomial with exactly three terms is called a trinomial. If it has one term, it's a monomial, and if it has two terms, it's a binomial. Any expression with more than three terms is still considered a polynomial.
Q3: What is the degree of the polynomial 2x^{3}  5x^{2} + 7x  9? (1 Mark)
(a) 2
(b) 3
(c) 4
(d) 1
Ans: (b)
The degree of a polynomial is the highest power of the variable present in the expression. In this case, the highest power of x is 3 (in the term 2x^3), so the degree of the polynomial is 3.
Q4: State whether the following statement is True or False: "A polynomial of degree 0 is called a constant polynomial." (1 Mark)
Ans: True
A polynomial of degree 0 is called a constant polynomial. It is a polynomial with no variable term, and the expression consists only of a constant value.
Q5: Identify the coefficients of the following polynomial: 3x^{2}  8x + 5. (1 Mark)
Ans: The coefficients are 3, 8, and 5.
The coefficients of a polynomial are the numerical values multiplied by each variable term. In this case, the coefficients are 3 (multiplied by x^2), 8 (multiplied by x), and 5 (constant term).
Q6: Find the value of 'k' in the polynomial x^{2}  kx + 9, if one of its zeros is 3. (2 Marks)
Ans: k = 6
If one of the zeros of a polynomial is 3, it means that when x = 3, the polynomial becomes equal to 0. Substituting x = 3 in the given polynomial and solving for 'k':
3^{2}  3k + 9 = 0
9  3k + 9 = 0
3k + 18 = 0
3k = 18
k = 18 / 3
k = 6
Q7: Factorize the polynomial completely: 2x^{3}  8x^{2} + 4x. (2 Marks)
Ans: 2x(x^{2}  4x + 2)
To factorize the polynomial completely, first, we take out the common factor '2x' from each term:
2x^{3}  8x^{2} + 4x = 2x(x^{2}  4x + 2)
The expression (x^{2}  4x + 2) cannot be further factorized using integers. So, the polynomial is completely factored as 2x(x^{2}  4x + 2).
Q8: If (x + 2) is a factor of the polynomial 2x^{3} + kx^{2}  4x + 16, find the value of 'k'. (2 Marks)
Ans: k = 10
If (x + 2) is a factor of the polynomial, it means that when x = 2, the polynomial becomes equal to 0. Substituting x = 2 in the given polynomial and solving for 'k':
2(2)^{3} + k(2)^{2}  4(2) + 16 = 0
16 + 4k + 8 + 16 = 0
4k + 8 = 0
4k = 8
k = 8 / 4
k = 2
Q9: If the sum of the zeroes of the polynomial x^{2}  kx + 15 is 5, find the value of 'k'. (3 Marks)
Ans: k = 10
The sum of the zeroes of the polynomial x^{2}  kx + 15 is given by the formula: Sum of zeroes = Coefficient of x / Coefficient of x^{2}.
In this case, the sum of zeroes is 5. So,
5 = (k) / 1
5 = k
Therefore, the value of k is 5.
Q10: The area of a rectangular garden is given by the polynomial 2x^{2 }+ 7x  15. Find the dimensions of the garden if its length is 2 units more than its breadth. (3 Marks)
Ans: Length = 5 units, Breadth = 3 units
Let the breadth of the garden be 'x' units. According to the question, the length of the garden is 2 units more than the breadth, which means the length is (x + 2) units.
The area of the rectangle is given by length × breadth. So, we have:
Area = (x + 2) × x = 2x^{2} + 2x
According to the problem, the area is also given by 2x^{2} + 7x  15.
Equating the two expressions for the area:
2x^{2} + 2x = 2x^{2} + 7x  15
2x = 7x  15
15 = 7x  2x
15 = 5x
x = 3
So, the breadth of the garden is 3 units, and the length is (3 + 2) = 5 units.
Q11: Find the value of 'p' for which (2p  1) is a factor of the polynomial 4p^{3}  6p^{2} + 3p  1. (3 Marks)
Ans: p = 1/2
If (2p  1) is a factor of the polynomial 4p^{3}  6p^{2 }+ 3p  1, it means that when we divide the polynomial by (2p  1), the remainder should be zero. Using the remainder theorem, we can find the value of 'p'.
Put (2p  1) = 0 to find the value of 'p':
2p  1 = 0
2p = 1
p = 1/2
Therefore, the value of 'p' is 1/2.
Q12: Explain the Remainder Theorem and how it can be used to find the remainder of a polynomial division. (5 Marks)
Ans: The Remainder Theorem states that if you divide a polynomial f(x) by (x  a), the remainder will be equal to f(a).
Mathematically, for a polynomial f(x) and a constant 'a,' when we divide f(x) by (x  a), the remainder 'R' is given by: R = f(a).
The Remainder Theorem is particularly useful when we want to find the remainder of a polynomial division without performing the long division or synthetic division. It allows us to find the value of the polynomial at a specific point 'a' without the need for a complete division process.
For example, if we want to find the remainder when dividing the polynomial f(x) = 2x^{3}  3x^{2} + 4x  5 by (x  2), we can use the Remainder Theorem. According to the theorem, the remainder will be f(2) = 2(2)^{3}  3(2)^{2} + 4(2)  5 = 8  12 + 8  5 = 1.
By using the Remainder Theorem, we can quickly evaluate the remainder of the division and save time compared to long division or synthetic division. This theorem is a fundamental concept used in various applications in algebra and calculus.
Q13: Explain the Factor Theorem and how it can be used to determine whether a given expression is a factor of a polynomial. (5 Marks)
Ans: The Factor Theorem states that if (x  a) is a factor of a polynomial f(x), then f(a) = 0.
Mathematically, for a polynomial f(x) and a constant 'a,' if (x  a) is a factor of f(x), then f(a) = 0.
The Factor Theorem provides a powerful tool for determining whether a given expression is a factor of a polynomial without performing the division process. If we can find a value 'a' such that f(a) equals zero, then it confirms that (x  a) is indeed a factor of f(x).
For example, consider the polynomial f(x) = x^{2}  4x + 4. To check whether (x  2) is a factor of f(x), we can use the Factor Theorem. If (x  2) is a factor, then f(2) should be equal to zero.
Evaluating f(2) = (2)^{2}  4(2) + 4 = 4  8 + 4 = 0
Since f(2) equals zero, the Factor Theorem confirms that (x  2) is indeed a factor of f(x).
The Factor Theorem is an essential concept used in solving polynomial equations, finding roots (zeros) of polynomials, and factorizing polynomials into their linear factors. It simplifies the process of factorization and helps in understanding the relationship between factors and zeros of a polynomial.
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