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
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