Mathematics
Identify which of the following expressions are polynomials. If so, write their degrees.
(i)
(ii) 8x2 - 3x + 6√x + 1
(iii) 5x2 - + 7
(iv) 9x2y2 - 3xy2 + 5x2y - 6x
(v) 6p4 - p3q2 + pq3 + q4
(vi) 4x5 - 7x5y + 3xy4 + 8y5
(vii) ab2 - + 5b2 + 6
Algebraic Expressions
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Answer
(i)
All powers of x are whole numbers.
The highest power of x is 4.
Yes, it is a polynomial. Degree = 4
(ii) 8x2 - 3x + 6√x + 1
The term 6√x can be written as 6x1/2, 1/2 is not a whole number.
∴ It is not a polynomial.
(iii) 5x2 - + 7
The term can be written as 2x-1, power of x is a negative integer.
∴ It is not a polynomial.
(iv) 9x2y2 - 3xy2 + 5x2y - 6x
All powers of x and y are whole numbers.
Degrees of terms:
9x2y2 → 2 + 2 = 4
3xy2 → 1 + 2 = 3
5x2y → 2 + 1 = 3
6x → 1
The highest degree is 4.
Yes, it is a polynomial. Degree = 4
(v) 6p4 - p3q2 + pq3 + q4
All powers of p and q are whole numbers.
Degrees of terms:
6p4 → 4
p3q2 → 3 + 2 = 5
pq3 → 1 + 3 = 4
q4 → 4
The highest degree is 5.
Yes, it is a polynomial. Degree = 5
(vi) 4x5 - 7x5y + 3xy4 + 8y5
All powers of x and y are whole numbers.
Degrees of terms:
4x5 → 5
7x5y → 5 + 1 = 6
3xy4 → 1 + 4 = 5
8y5 → 5
The highest degree is 6.
Yes, it is a polynomial. Degree = 6
(vii) ab2 - + 5b2 + 6
The term means 7a-2, power of a is a negative integer.
∴ It is not a polynomial.
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Related Questions
Identify like terms in the following:
(i) a2, b2, -2a2, c2, 4a
(ii) 3x, 4xy, -yz, zy
(iii) -2xy2, x2y, 5y2x, x2z
(iv) abc, ab2c, acb2, c2ab, b2ac, a2bc, cab2
Generate algebraic expressions for the following:
(i) The product of a and b added to their sum.
(ii) The quotient of x by 8 is multiplied by y.
(iii) Thrice x added to y squared.
(iv) One-third of x multiplied by the sum of p and q.
(v) The number to be added to x + y to make it equal to p.
(vi) From a rod (p + q) units in length, n equal pieces are cut. Find the length of each piece.
(vii) Five times m subtracted from the sum of p and thrice x.
(viii) The number obtained when m times the difference of x and y is subtracted from n times the sum of x and y.
(ix) The sum of 7 numbers each equal to p.
(x) The product of three numbers a, b and c subtracted from the sum of x and y.
Add the following expressions:
(i) 2x2, -5x2, -x2, 6x2
(ii) x2 - 2xy + 3y2, 5y2 + 3xy - 6x2
(iii) 2x + 9y - 7z, 3y + z - 3x, 2z - 4y - x
(iv) 2ab + 3bc - 5ca, 4bc - 3ab + 7ca, 2ca - ab - 5bc
(v) 3x3 + 2x2 - 6x + 3, 2x3 - 3x2 - x - 4, 1 + 2x - 3x2 - 4x3
(vi) 3n2 + 5mn - 6m2, 2m2 - 3mn - 4n2, 2mn - 3m2 - 7n2
(vii) 3z3 - z2 + 5, 1 - 2z + z2, 3 + 2z - z3
Simplify:
(i) 5x + 3y - 8z + 2y - 3x + 5z + z - 7y - 2x
(ii) 4x3 - 2x2 + 5x - 1 + 8x + x2 - 6x3 + 7 - 6x + 3 - 3x2 - x3
(iii) 2x2 + 3xy - 3y2 + x2 - xy + y2
(iv) 2 - 3z2 + 5yz + 7y2 - 8 + z2 - 6yz - 9y2 + 1 - 2z2 - 2yz - y2
(v) 2m - 3n + 5p + 2m + n - 2p - 3m - 4n + p