Question

If (x-2)and(x-1) are the factors of polynomial x
${x}^{3}$
+10
${x}^{2}$
+ax-b then find the values of a and b​

Answer 1

Given -

(x - 2) and (x - 1) are factors of polynomial p(x) = x³ + 10x² + ax - b

To Find -

• Value of a and b

As we know that :-

If (x - 2) is a factor then x = 2 is the zero of the polynomial

And

If (x - 1) is a factor then x = 1 is the zero of the polynomial

Now,

p(x) = x³ + 10x² + ax - b

→ p(2) = (2)³ + 10(2)² + a(2) - b

→ 8 + 40 + 2a - b = 0

→ 2a - b = -48 ........ (i)

And

p(1) = (1)³ + 10(1)² + a(1) - b

→ 1 + 10 + a - b = 0

→ a - b = -11 ........ (ii)

Now, By solving (i) and (ii), we get :

→ 2a - b = -48

a - b = -11

(-) (+) (+)

___________

→ a = -37

And

Now, Substituting the value of a on a - b = -11, we get :

→ a - b = -11

→ b = a + 11

→ b = -37 + 11

→ b = -26

Hence,

The value of a is -37 and b is -26

Verification :-

→ x³ + 10x² - 37x + 26

→ x³ - 2x² + 12x² - 24x - 13x + 26

→ x²(x - 2) + 12x(x - 2) - 13(x - 2)

→ (x - 2)(x² + 12x - 13)

→ (x - 2)(x² - x + 13x - 13)

→ (x - 2)[x(x - 1) + 13(x - 1)]

→ (x - 2)(x + 13)(x - 1)

Here, The factors comes same as given in the question.

It shows that our answer is absolutely correct.

Answer 2

$\large{\underline{\bf{\purple{Given:-}}}}$

• ✦ p(x) = x³ + 10x² + ax - b
• and two factors are given as (x - 2) and (x -1)

$\large{\underline{\bf{\purple{To\:Find:-}}}}$

✦ we need to find the value of a and b.

$\huge{\underline{\bf{\red{Solution:-}}}}$

p(x) = x³ + 10x² + ax - b

factors are (x - 2) and (x - 1)

So, x = 2 and x = 1

putting value of x =2 in equation :-

x³ + 10x² + ax - b

$\longmapsto \rm\: {(2)}^{3} + 10 \times {(2)}^{2} + a \times 2 - b\: \\ \\\longmapsto \rm\:8 + 40 + 2a - b \\ \\\longmapsto \rm\:2a - b = - 48........(i)$

Now putting value of x = 1 in equation:-

x³ + 10x² + ax - b

$\longmapsto \rm\:\: {(1)}^{3} + 10 \times {1}^{2} + a \times 1 - b \\ \\ \longmapsto \rm\:\:1 + 10 + a - b \\ \\ \longmapsto \rm\:\:a - b = -11.......(ii)$

$\rm\underbrace{ \green{Solving\:(i)\:and\:(ii)\:By \: elimination \: method }} \:$

$\rm\:\:2a - b = - 48 \\ \rm\:\:a \: \: - b = - 11 \\\rm\:\: - \: \: \: \: \: + \: \: \: \: \: + \\ \bf\underline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: } \\ \rm\:\:\pink{a \: \: \: \: \: \: \: \: \: \: \: \: \: = - 37 }\\ \\ \rm\:putting \: value \: of \: a \: in \: eq.\:(ii) \\ \\ \longmapsto \rm \:\:\:a - b = - 11 \\ \\ \longmapsto \rm \: \: - 37 - b = - 11 \\ \\ \longmapsto \rm \: - b = - 11 + 37 \\ \\\longmapsto \rm - b = 26 \\ \\ \longmapsto \bf \: \pink{ b = - 26}\:$

Hence

• a = -37
• b = -26

━━━━━━━━━━━━━━━━━━━━━━━━━