A polynomial ax3 - bx2 - cx + 3 is divisible byx2 - 1. If 2x−3 is a factor, Find the values of a, b and c

Question

ScanSolve AI Tutor · Accepted Answer

Ili kutatua swali hili, tutatumia Theorem ya Mabaki (Remainder Theorem).

Tumia masharti ya kugawanyika.
Polynomial $P(x) = ax^3 - bx^2 - cx + 3$ inagawanyika na $x^2 - 1$ . Hii inamaanisha inagawanyika na $(x-1)$ na $(x+1)$ . Kwa hiyo, $P(1) = 0$ na $P(-1) = 0$ .
Pia, $2x - 3$ ni kigezo, hivyo $P\left(\frac{3}{2}\right) = 0$ .
Unda milinganyo kutoka kwa masharti.
Kutoka $P(1) = 0$ :
$a(1)^3 - b(1)^2 - c(1) + 3 = 0$ $a - b - c + 3 = 0 \quad \text{(Mlinganyo 1)}$
Kutoka $P(-1) = 0$ :
$a(-1)^3 - b(-1)^2 - c(-1) + 3 = 0$ $-a - b + c + 3 = 0 \quad \text{(Mlinganyo 2)}$
Kutoka $P\left(\frac{3}{2}\right) = 0$ :
$a\left(\frac{3}{2}\right)^3 - b\left(\frac{3}{2}\right)^2 - c\left(\frac{3}{2}\right) + 3 = 0$ $a\left(\frac{27}{8}\right) - b\left(\frac{9}{4}\right) - c\left(\frac{3}{2}\right) + 3 = 0$
Kuzidisha mlinganyo mzima kwa 8 ili kuondoa sehemu:
$27a - 18b - 12c + 24 = 0$
Kugawanya mlinganyo mzima kwa 3:
$9a - 6b - 4c + 8 = 0 \quad \text{(Mlinganyo 3)}$
Tatua mfumo wa milinganyo.
Tutaongeza Mlinganyo 1 na Mlinganyo 2:
$(a - b - c) + (-a - b + c) = -3 + (-3)$ $-2b = -6$ $b = 3$
Badilisha $b=3$ katika Mlinganyo 1:
$a - 3 - c + 3 = 0$ $a - c = 0$ $a = c$
Badilisha $b=3$ na $a=c$ katika Mlinganyo 3:
$9a - 6(3) - 4a + 8 = 0$ $9a - 18 - 4a + 8 = 0$ $5a - 10 = 0$ $5a = 10$ $a = 2$
Kwa kuwa $a=c$ , basi $c=2$ .

Kwa hiyo, maadili ni $a=2$ , $b=3$ , na $c=2$ .

Maadili ya $a$ , $b$ , na $c$ ni:
$a = \boxed{2}$
$b = \boxed{3}$
$c = \boxed{2}$

A polynomial ax3 - bx2 - cx + 3 is divisible byx2 - 1. If 2x-3 is a factor, Find the values of a, b and c