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The polynomial 2x^3–ax^2+5bx+4b has a factor x–2 and, when divided by x+1, a remainder of –15 is obtained. Find the values of a and b. ?
With these values of a and b, factorize the polynomial completely. Hence, or otherwise, find all the real factors of 2x^6–11x^4+10x^2+8
WHAT IS THE COMPLETE FACTORIZED FORM OF THE POLYNOMIAL?
2 Answers
- ?Lv 61 year ago
f(x) = 2x³-ax²+5bx+4b
x-2 = 0 → x – 2
f(2) = 2(2)³ - a(2)² + 5b(2) + 4b
= 16-4a+10b+4b
f(2) = 16-4a+14b ……….(1)
x+1 = 0 → x = -1
f(-1) = -2-a-5b+4b = -15
→ -2-a-b = -15
a+b = 13………….(2)
equation (1) and (2)
→ a = 13-b……sub in equation (1)
16-4(13-b)+14b = 0
16-52+4b+14b = 0
18b – 36 + 18b = 0
36b = 36 → b = 1
a = 13-1 → a = 12
Source(s): http://myrank.co.in/ - stanschimLv 71 year ago
That (x-2) is factor implies that f(2) = 0 =16-4a+14b.
That (x+1) divides the polynomial with a remainder of -15 implies that
f(-1) = -15 = -2-a-b.
Solving these two equations in two unknowns gives a = 11 and b = 2.
