The terms of a polynomial may be written in any order. However, polynomials that contain only one variable are usually written in standard form.
The standard form of a polynomial that contains one variable is written with the terms in order from greatest degree to least degree. When written in standard form, the coefficient of the first term is called the leading coefficient.
Examples :
(i) 8x4 + 4x3 - 7x2 - 9x + 6
(ii) -y4 + 4y3 6y2 - 3y + 5
Question 1 :
Rewrite each the following polynomials in standard form.
(i) x - 9 + √7x3 + 6x2
Solution :
Standard form of the given polynomial is
√7x3 + 6x2 + x - 9
(ii) √2x2 - (7/2)x4 + x - 5x3
Solution :
Standard form of the given polynomial is
- (7/2)x4 - 5x3 + √2x2 + x
(iii) 7x3 - (6/5)x2 + 4x - 1
Solution :
The given polynomial is already in standard form.
(iv) y2 - √5y3 - 11 - (7/3)y + 9y4
Solution :
Standard form of the given polynomial is
= 9y4- √5y3 + y2 - (7/3)y - 11
Question 2 :
Add the following polynomials and write the resultant polynomials in standard form.
(i) p(x) = 6x2 - 7x + 2 and q(x) = 6x3 - 7x + 15
Solution :
p(x) + q(x) = (6x2 - 7x + 2) + (6x3 - 7x + 15)
= 6x2 - 7x + 2 + 6x3 - 7x + 15
= 6x3 + 6x2 - 7x - 7x + 2 + 15
= 6x3 + 6x2 - 14x + 17
(ii) h(x) = 7x3 - 6x + 1, f(x) = 7x2 + 17x - 9
Solution :
h(x) + f(x) = (7x3 - 6x + 1) + (7x2 + 17x - 9)
= 7x3 + 7x2- 6x + 17x + 1 - 9
= 7x3 + 7x2 + 11x - 8
(iii) f(x) = 16x4 - 5x2 + 9, g(x) = -6x3 + 7x - 15
Solution :
f(x) + g(x) = (16x4 - 5x2 + 9) + (-6x3 + 7x - 15)
= 16x4- 6x3 - 5x2 + 7x + 9 - 15
= 16x4- 6x3 - 5x2 + 7x - 6
Question 3 :
If m = 4n + 5 and p = 3n + 6, which of the following expresses n in terms of m and p?
(A) m + 2p - 3 (B) m - p + 1 (C) 3m - 2p + 4
(D) 4m + p + 1 (E) 2m - 5p
Solution :
m = 4n + 5 and p = 3n + 6
Writing 3n as 4n - 1n and 6 as 5 + 1,
p = 4n - n + 5 + 1
p = (4n + 5) - n + 1
p = m - n + 1
So, option B is correct.
Question 4 :
Which expression is equivalent to
(x2 + 11)2 + (x + 5)(x - 5) ?
A) x4 + 23x2 - 14 B) x4 + 23x2 + 96
C) x4 + 12x2 + 121 D) x4 + x2 + 146
Solution :
= (x2 + 11)2 + (x + 5)(x - 5)
= x4 + 2x2 (11) + 112 + x2 - 52
= x4 + 22x2 + 121 + x2 - 25
= x4 + 23x2 + 96
So, option B is correct.
Question 5 :
x2 y - 3y2 + 5xy2 - (-x2 y + 3xy2 - 3y2)
Solution :
= x2 y - 3y2 + 5xy2 - (-x2 y + 3xy2 - 3y2)
= x2 y - 3y2 + 5xy2 + x2 y - 3xy2 + 3y2
= x2 y + x2 y - 3y2 + 3y2 + 5xy2 - 3xy2
= 2x2 y + 2xy2
Question 6 :
If x + y = 10, what is the sum of (x + y)2 and (x + y) ?
A) 90 B) 100 C) 110 D) 200 E) 210
Solution :
Given that, x + y = 10
= (x + y)2 + (x + y)
= 102 + 10
= 100 + 10
= 110
So, the answer is option C.
Question 7 :
(x + 5) + (2x - 3)
Which of the following in equivalent to the given expression.
A) 3x - 2 B) 3x + 2 C) 3x - 8 D) 3x + 8
Solution :
= (x + 5) + (2x - 3)
= x + 5 + 2x - 3
= x + 2x + 5 - 3
= 3x + 2
So, option B is correct.
Question 8 :
What is the distance around the rectangle if the length is 3x2 + 6x – 10 and the width is 3x + 5?
Solution :
Length of the rectangle = 3x2 + 6x – 10
Width of the rectangle = 3 x + 5
Perimeter of the rectangle = 2(length + width)
= 2 (3x2 + 6x – 10 + 3x + 5)
= 2 (3x2 + 6x + 3x – 10 + 5)
= 2 (3x2 + 9x – 5)
Distributing 2, we get
= 6x2 + 18x – 10
So, the required perimeter of the rectangle is
6x2 + 18x – 10
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