Question

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Answer 1

Question:

Without actual division, prove that 2x⁴ - 5x³ + 2x² - x + 2 is divisible by x² - 3x + 2

Answer:

Step-by-step explanation:

$\Large{\underline{\underline{\bf{Given:}}}}$

• f (x) = 2x⁴ - 5x³ + 2x² - x + 2
• g (x) =  x² - 3x + 2

$\Large{\underline{\underline{\bf{To\:Prove:}}}}$

• f(x) is divisible by g(x)

$\Large{\underline{\underline{\bf{Proof:}}}}$

➛ First factorize g(x)

➛ g(x) = x² - 3x + 2

➛ Factorizing by splitting the middle term,

   x² - 2x -x + 2

   x (x - 2) -1 (x - 2)

  (x - 2) (x - 1)

➛ Hence (x - 2) and (x - 1) are the factors of g(x)

➛ x = 2, x = 1

➛ Substitute the value of x in f(x)

➛ f(2) = 2 × 2⁴ - 5 × 2³ + 2 × 2² - 2 + 2

   f(2) = 32 - 40 + 8 - 2 + 2

   f(2) = -8 + 8

   f (2) = 0

➛ f(1) = 2 × 1³ - 5 × 1³ + 2 × 1² - 1 + 2

   f(1) = 2 - 5 + 2 - 1 + 2

   f(1) = -3 + 3

   f(1) = 0

➛ Hence 1 and 2 are the zeros of the polynomial f(x).

➛ Hence f(x) is divisible by g(x)

➛ That is,

    2x⁴ - 5x³ + 2x² - x + 2 is divisible by x² - 3x + 2

$\Large{\underline{\underline{\bf{Notes:}}}}$

➡ A quadratic polynomial can be factorized by

• Using the quadratic formula
• Splitting the middle term
• Completing the square method

Answer 2

Answer :-

2x⁴ - 5x³ + 2x² - x + 2 is divisible by x² - 3x + 2.

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★Concept :-

Here, the concept of Factor Theorem is used. According to this, if we need to find the divisibility of two numbers, we use this to substitute the value of the variable.

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★Solution :-

Given,

• Dividend = 2x⁴ - 5x³ + 2x² - x + 2 {p(x)}

• Divisor = x² - 3x + 2 {g(x)}

Here, if we find the zeroes of the Divisor or simply to say the value of x, we can apply in the Dividend by using Factor Theorem.

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To prove :- 2x⁴ - 5x³ + 2x² - x + 2 is divisible by x² - 3x + 2

________________________________

✒ g(x) = x² - 3x + 2

By using the method of splitting the middle term, we get,

✒ x² - 2x - 1x + 2 = 0

✒ x{x - 2} - 1{x - 2} = 0

✒ (x - 1)(x - 2) = 0

Here either (x - 1) = 0 , or (x - 2) = 0

So,

▶ (x - 1) = 0 or (x - 2) = 0

▶ x = 1 or x = 2

So, x = 1 , 2

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Now, we are given,

✏ p(x) = 2x⁴ - 5x³ + 2x² - x + 2

We already got the value of x that is 1 or 2.

So let us choose anyone. (I am using here 2 , you can use 1 also)

✏ p(2) = 2(2)⁴ - 5(2)³ + 2(2)² - 2 + 2

Now by applying Factor Theorem here, we get,

✏ 2(2)⁴ - 5(2)³ + 2(2)² - 2 + 2 = 0

✏ 2(16) - 5(8) + 2(2)² - 2 + 2 = 0

✏ 32 - 40 + 8 = 0

✏ 40 - 40 = 0

✏ 0 = 0

Clearly, LHS = RHS

So, we find that, x is the factor of Dividend also and Divisor also, by factor theorem, since both sides are zero.

Finally we get our answer, that

2x⁴ - 5x³ + 2x² - x + 2 is divisible by x² - 3x + 2

Hence proved.

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★ More to know :-

• Factor Theorem states that when two polynomials are in relation of division with each other, the factor of one polynomial may or may not be the factor of the second. This can be used to prove the divisibility with actual division.

• Remainder Theorem states that when the remainder is obtained by dividing Dividend by a Divisor, the remainder so obtained is surely the factor of quotient obtained.

• Polynomial is the equation which has both variable terms and constant term. On the basis zeroes, polynomials are :-

Linear Polynomial - One Zero

Quadratic Polynomial - Two Zeroes

Cubic Polynomial - Three Zeroes

Bi - Quadratic Polynomial - More than three zeroes