Question

SHOW THAT 2x + 7 IS A FACTOR OF 2X^3 + 5X^2 - 11X - 14. HENCE FACTORIZE THE GIVEN EXPRESSION COMPLETELY BY FACTOR THEOREM.
(BOARD QUESTIONS CLASS 10)

Answer 1

Answer:

• (2x + 7)(x - 2)(x + 1) are the factors.

Step-by-step explanation:

Given :

f(x) = 2x³ + 5x² - 11x - 14.

2x + 7 is a factor of f(x).

To find :

Factors of f(x) by factor theorem.

Solution :

Let f(x) be 2x³ + 5x² - 11x - 14.

Now, (2x + 7) = 0, x = -7/2

[∵ Factor theorem]

∴ f(-7/2) will be:

⇒ 2(-7/2)³ + 5(-7/2)² -11(-7/2) - 14

⇒ 2(-343/8) + 5(49/4) + 77/2 - 14

⇒ -343 + 245 + 154 - 56/4

⇒ 4/4

⇒ 0

∴ (2x + 7) is a factor of f(x).

2x + 7(2x³ + 5x² - 11x - 14(x² - x - 2

          2x³ + 7x²

          _______________

          -2x² - 11x - 14

          -2x² - 7x

          _______________

                  -4x - 14

                  -4x - 14

          _______________

                          0

∴ On dividing f(x) by (2x + 1), we get x² - x - 2 as quotient.

∴ f(x) = 2x³ + 5x² - 11x -14

=> Dividend = Divisor × Quotient + Remainder

⇒ (2x + 7)(x² - x - 2) + 0

⇒ (2x + 7)(x² - 2x + x - 2)

⇒ (2x + 7)[x(x - 2) + 1(x - 2)]

⇒ (2x + 7)(x - 2)(x + 1)

∴ Factors of f(x) are (2x + 7), (x - 2), (x + 1).

Learn more :

Factor theorem: If f(x) is the polynomial and a is a real number. Then (x - a) is factor of f(x), f(a) = 0.

Remainder theorem: If f(x) is the polynomial and is divided by (x - a), then the remainder is f(a).

Answer 2

ǫᴜᴇsᴛɪᴏɴ →

SHOW THAT 2x + 7 IS A FACTOR OF 2X^3 + 5X^2 - 11X - 14. HENCE FACTORIZE THE GIVEN EXPRESSION COMPLETELY BY FACTOR THEOREM.

ᴀɴsᴡᴇʀ →

we know that, if (x + a) is a factor of any polynomial p(x) , than, at p(-a) , value of p(x) is equal to zero.

So,

$\longmapsto\tt{ 2x + 7 = 0}$

$\longmapsto\tt{ 2x = (-7)}$

$\longmapsto\tt{x = (-7/2)}$

Putting x = (-7/2), we gets,

$\longmapsto\tt{p(x) = 2x^3 + 5x^2 - 11x - 14}$

$\longmapsto\tt{\dfrac{-7}{2} = 2} \: \: \longmapsto\tt{\dfrac{-7}{2³} + x 5}$

$\longmapsto\tt{\dfrac{-7}{2²}- 11}$

(-7/2)² - 11(-7/2) - 14

→ p(-7/2) = 2(-343/8) + 5(49/4) + (77/2) - 14

→ p(-7/2) = (-686/8) + (245/4) + (77/2) - (14/1)

→ p(-7/2) = (-686 + 490 + 308 - 112)/8

→ p(-7/2) = (798 - 798) / 8

→ p(-7/2) = 0 .

Therefore, (2x + 7) is a factor of 2x^3+5x^2-11x-14 .

Now, Dividing the given polynomial by (2x + 7) , we get,

$\longmapsto\tt{2x+7 )2x³ + 5x² - 11x - 14(x² - x - 2}$

$\longmapsto\tt{-2x³ + 7x²}$

$\longmapsto\tt{-2x² - 11x}$

$\longmapsto\tt{-2x² - 7x}$

$\longmapsto\tt{-4x - 14}$

$\longmapsto\tt{-4x - 14}$

__0___

Quotient = (x² - x - 2)

using common factor theorem we get,

$\longmapsto\tt{ x² - 2x + x - 2}$

$\longmapsto\tt{x (x - 2) + 1(x - 2)}$

$\longmapsto\tt{ (x - 2)(x + 1)}$

Hence, the factorise form of the given polynomial is (x + 2)(x - 2)(2x + 7) .