Question
$(x-2)$ is a factor of the expression $x^3+a x^2+b x+6$. When this expression is divided by $(x-3)$, it leaves the remainder 3 . Find the values of $a$ and $b$.
✓
Answer
As $x – 2$ is a factor of
$f(x) = x^3 + ax^2 + bx + 6$
$\therefore f(2) = 0$
$\therefore (2)^3 + a(2)^2 + b(2) + 6 = 0$
$\Rightarrow 8 + 4a + 2b + 6 = 0$
$\Rightarrow 4a + 2b = –14$
$\Rightarrow 2a + b = –7 ....(i)$
as on dividing f(x) by$ x – 3$
remainder $= 3$
$\therefore f(3) = 3$
$\therefore (3)^3 + a(3)^2 + b(3) + 6 = 3$
$\Rightarrow 27 + 9a + 3b + 6 = 3$
$\Rightarrow 9a + 3b = –30$
$\Rightarrow 3a + b = –10 ....(ii)$
Solving simultaneously equation (i) and (ii),
$\therefore 2a + b = –7$
$3a + b – 10$
Subtracting $– – +$
$–a = 3 $
$a = –3$
Subtracting value of a in equation (i)
$2(–3) + b = –7$
$\therefore –6 + b = –7$
$\therefore b = –1$
$\therefore a = –3, b = –1.$
$f(x) = x^3 + ax^2 + bx + 6$
$\therefore f(2) = 0$
$\therefore (2)^3 + a(2)^2 + b(2) + 6 = 0$
$\Rightarrow 8 + 4a + 2b + 6 = 0$
$\Rightarrow 4a + 2b = –14$
$\Rightarrow 2a + b = –7 ....(i)$
as on dividing f(x) by$ x – 3$
remainder $= 3$
$\therefore f(3) = 3$
$\therefore (3)^3 + a(3)^2 + b(3) + 6 = 3$
$\Rightarrow 27 + 9a + 3b + 6 = 3$
$\Rightarrow 9a + 3b = –30$
$\Rightarrow 3a + b = –10 ....(ii)$
Solving simultaneously equation (i) and (ii),
$\therefore 2a + b = –7$
$3a + b – 10$
Subtracting $– – +$
$–a = 3 $
$a = –3$
Subtracting value of a in equation (i)
$2(–3) + b = –7$
$\therefore –6 + b = –7$
$\therefore b = –1$
$\therefore a = –3, b = –1.$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
The fifth, eight and eleventh terms of a geometric progression are p, q and r respectively. show that : $q^2 = pr$.
→In the figure, given alongside, CP bisects angle ACB. Show that DP bisects angle ADB.
The given figure shows a circle with centre O such that chord RS is parallel to chord QT, angle PRT = 20° and angle POQ = 100°. Calculate:
(iv) angle STR
(iv) angle STR
The side AB of a square ABCD is parallel to x-axis. Find the slopes of all its sides. Also, find:
i. the slope of the diagonal AC.
ii. the slope of the diagonal BD.
i. the slope of the diagonal AC.
ii. the slope of the diagonal BD.
Calculate the percentage income in the following investments:
Rs 36,250 in a Rs 125 share paying 8% and available at a premium of Rs 20.
Rs 36,250 in a Rs 125 share paying 8% and available at a premium of Rs 20.
The marked price of an article is Rs. 6000. A wholesaler sells it to a dealer at 20% discount. The dealer further sells the article to a customer at a discount of 10% on the marked price. If the rate of GST at each stage is 18%, find the amount of tax (under GST) paid by the dealer to the government.
If $A: B=\frac{1}{4}: \frac{1}{5}$ and $B: C=\frac{1}{7}: \frac{1}{6}$, find $A: B: C$.
→In what ratio does the point $C \left(\frac{5}{5}, \frac{11}{5}\right)$ divide the line segment joining the points $A (3,5)$ and $B (-3,-2)$ ?
→Find the equation of a line passing through the point $(2, 3)$ and having the x-intercept of $4$ units.
→The 4th, 9th and last terms of a GP are 10, 320 and 2560 respectively, find the first term and the number of terms in the GP.