Sum Difference Product and Quotient of Functions :
Here we are going to see, how to find sum, difference product and quotient of functions.
If p and q are nonzero polynomials, then
deg(p + q) ≤ maximum{deg p, deg q}
and
deg(p − q) ≤ maximum{deg p, deg q}.
Degree of the product of two polynomials If p and q are nonzero polynomials, then
deg(pq) = deg p + deg q.
Question 1 :
Suppose
p(x) = x2 + 5x + 2, q(x) = 2x3 − 3x + 1, s(x) = 4x3 − 2
write the indicated expression as a sum of terms, each of which is a constant times a power of x.
(i) (4p + 5q)(x) (ii) (pq)(x) (iii) (ps)(x) (iv) (p(x))2
(v) (q(x))2 (vi) (p(x))2s(x)
Solution :
(i) (4p + 5q)(x) = 4 p(x) + 5 q(x)
= 4(x2 + 5x + 2) + 5(2x3 − 3x + 1)
= 4x2 + 20x + 8 + 10x3 − 15x + 5
= 10x3 + 4x2 − 15x + 20x + 5 + 8
= 10x3 + 4x2 + 5x + 13
(ii) (pq)(x)
p(x) = x2 + 5x + 2, q(x) = 2x3 − 3x + 1
(pq)(x) = p(x) ⋅ q(x)
= (x2 + 5x + 2) (2x3 − 3x + 1)
= x2 (2x3 − 3x + 1) + 5x (2x3 − 3x + 1) + 2 (2x3 − 3x + 1)
= 2x5 - 3x3 + x2 + 10x4 - 15x2 + 5x + 4x3 - 6x + 2
= 2x5 + 10x4 - 3x3 + 4x3 + x2 - 15x2 + 5x - 6x + 2
= 2x5 + 10x4 + x3 - 14x2 x + 2
(iii) (ps)(x)
= p(x) ⋅ s(x)
p(x) = x2 + 5x + 2 s(x) = 4x3 − 2
= (x2 + 5x + 2) ⋅ (4x3 − 2)
= x2(4x3 − 2) + 5x(4x3 − 2) + 2(4x3 − 2)
= 4x5 - 2x2 + 20x4 - 10x + 8x3 - 4
= 4x5 + 20x4 + 8x3- 2x2 - 10x - 4
(iv) (p(x))2
= (x2 + 5x + 2)2
= (x2)2 + (5x)2 + 22 + 2 x2(5x) + 2(5x) 2 + 2 (2)x2
= x4 + 25x2 + 4 + 10x3 + 20x + 4x2
= x4 + 10x3 + 29x2 + 20x + 4
(v) (q(x))2
= (2x3 − 3x + 1)2
= (2x3)2 + (-3x)2 + 12 + 2 (2x3)(-3x) + 2(-3x) 1 + 2(2x3)
= 4x6 + 9x2 + 1 - 12x4 - 6x + 4x3
= 4x6 - 12x4 + 4x3 + 9x2 - 6x + 1
(vi) (p(x))2s(x)
= (x4 + 10x3 + 29x2 + 20x + 4)(4x3 − 2)
= x4(4x3−2)+10x3(4x3−2)+29x2(4x3−2)+20x(4x3−2)+ 4(4x3−2)
= 4x7−2x4+40x6−20x3 + 116x5 - 58x2 + 80x4 - 40x + 16x3 - 8
= 4x7+40x6+ 116x5 −2x4 + 80x4 + 16x3−20x3 - 58x2 - 40x - 8
= 4x7+40x6+ 116x5 + 78x4 - 4x3 - 58x2 - 40x - 8
After having gone through the stuff given above, we hope that the students would have understood "Sum Difference Product and Quotient of Functions".
Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.
Kindly mail your feedback to v4formath@gmail.com
We always appreciate your feedback.
©All rights reserved. onlinemath4all.com
Apr 26, 24 09:20 PM
Apr 26, 24 12:39 PM
Apr 26, 24 01:51 AM