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EXERCISE 8.1 PAGE NO: 8.2
1. Write the degree of each of the following polynomials:
(i) 2x3 + 5x2 – 7
(ii) 5x2 – 3x + 2
(iii) 2x + x2 – 8
(iv) 1/2y7 – 12y6 + 48y5 – 10
(v) 3x3 + 1
(vi) 5
(vii) 20x3 + 12x2y2 – 10y2 + 20
Solution:
(i) 2x3 + 5x2 – 7
We know that in a polynomial, degree is the highest power of the variable.
The degree of the polynomial, 2x3 + 5x2 – 7 is 3.
(ii) 5x2 – 3x + 2
The degree of the polynomial, 5x2 – 3x + 2 is 2.
(iii) 2x + x2 – 8
The degree of the polynomial, 2x + x2 – 8 is 2.
(iv) 1/2y7 – 12y6 + 48y5 – 10
The degree of the polynomial, 1/2y7 – 12y6 + 48y5 – 10 is 7.
(v) 3x3 + 1
The degree of the polynomial, 3x3 + 1 is 3
(vi) 5
The degree of the polynomial, 5 is 0 (since 5 is a constant number).
(vii) 20x3 + 12x2y2 – 10y2 + 20
The degree of the polynomial, 20x3 + 12x2y2 – 10y2 + 20 is 4.
2. Which of the following expressions are not polynomials?
(i) x2 + 2x-2
(ii) √(ax) + x2 – x3
(iii) 3y3 – √5y + 9
(iv) ax1/2 + ax + 9x2 + 4
(v) 3x-3 + 2x-1 + 4x + 5
Solution:
(i) x2 + 2x-2
The given expression is not a polynomial.
Because a polynomial does not contain any negative powers or fractions.
(ii) √(ax) + x2 – x3
The given expression is a polynomial.
Because the polynomial has positive powers.
(iii) 3y3 – √5y + 9
The given expression is a polynomial.
Because the polynomial has positive powers.
(iv) ax1/2 + ax + 9x2 + 4
The given expression is not a polynomial.
Because a polynomial does not contain any negative powers or fractions.
(v) 3x-3 + 2x-1 + 4x + 5
The given expression is not a polynomial.
Because a polynomial does not contain any negative powers or fractions.
3. Write each of the following polynomials in the standard from. Also, write their degree:
(i) x2 + 3 + 6x + 5x4
(ii) a2 + 4 + 5a6
(iii) (x3 – 1) (x3 – 4)
(iv) (y3 – 2) (y3 + 11)
(v) (a3 – 3/8) (a3 + 16/17)
(vi) (a + 3/4) (a + 4/3)
Solution:
(i) x2 + 3 + 6x + 5x4
The standard form of the polynomial is written in either increasing or decreasing order of their powers.
3 + 6x + x2 + 5x4 or 5x4 + x2 + 6x + 3
The degree of the given polynomial is 4.
(ii) a2 + 4 + 5a6
The standard form of the polynomial is written in either increasing or decreasing order of their powers.
4 + a2 + 5a6 or 5a6 + a2 + 4
The degree of the given polynomial is 6.
(iii) (x3 – 1) (x3 – 4)
x6 – 4x3 – x3 + 4
x6 – 5x3 + 4
The standard form of the polynomial is written in either increasing or decreasing order of their powers.
x6 – 5x3 + 4 or 4 – 5x3 + x6
The degree of the given polynomial is 6.
(iv) (y3 – 2) (y3 + 11)
y6 + 11y3 – 2y3 – 22
y6 + 9y3 – 22
The standard form of the polynomial is written in either increasing or decreasing order of their powers.
y6 + 9y3 – 22 or -22 + 9y3 + y6
The degree of the given polynomial is 6.
(v) (a3 – 3/8) (a3 + 16/17)
a6 + 16a3/17 – 3a3/8 – 6/17
a6 + 77/136a3 – 48/136
The standard form of the polynomial is written in either increasing or decreasing order of their powers.
a6 + 77/136a3 – 48/136 or -48/136 + 77/136a3 + a6
The degree of the given polynomial is 6.
(vi) (a + 3/4) (a + 4/3)
a2 + 4a/3 + 3a/4 + 1
a2 + 25a/12 + 1
The standard form of the polynomial is written in either increasing or decreasing order of their powers.
a2 + 25a/12 + 1 or 1 + 25a/12 + a2
The degree of the given polynomial is 2.
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