Question

find the factor of the polynomial x^3-5x^2+8x-4 divide the polynomial with the factor and verify the division rule ​

Answer 1

Answer:

Step-by-step explanation:

Use remainder factor theorem

x-1=0

x=1

put the value

x3-8x2+19x-12=0

LHS=

1-8+19-12

=0=RHS

Hence x-1 is a factor of the given expression

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Answer 2

Note:

★ The possible values of the variable for which the polynomial becomes zero are called its zeros .

★ A cubic polynomial can have atmost three zeros .

★ If p(a) = 0 , then x = a is a zero of polynomial p(x) and hence (x - a) is a factor of the polynomial p(x) .

★ If x - a is a factor of the polynomial p(x) then the remainder obtained on dividing p(x) by (x - a) is zero .

Solution:

Here,

The given cubic polynomial is ;

x³ - 5x² + 8x - 4 .

By hit and trial , we get that the polynomial becomes zero at x = 1 .

If x = 1 , then ;

x - 1 = 0 .

Thus,

(x - 1) is a factor of the given cubic polynomial .

Now,

To get another factor let's divide the given polynomial by (x - 1)

x - 1 ) x³ - 5x² + 8x - 4 ( x² - 4x + 4

x³ - x²

– +

- 4x² + 8x

- 4x² + 4x

+ –

4x - 4

4x - 4

– +

× ×

Here,

Dividend = x³ - 5x² + 8x - 4

Divisor = x - 1

Quotient = x² - 4x + 4

Remainder = 0

Also,

We know that ;

Dividend = Divisor×Quotient + Remainder

Now,

=> LHS = Dividend

=> LHS = x³ - 5x² + 8x - 4

Also,

=> RHS = Divisor × Quotient + Remainder

=> RHS = (x² - 4x + 4) × (x - 1)

=> RHS = x³ - x² - 4x² + 4x + 4x - 4

=> RHS = x³ - 5x² + 8x - 4

Clearly,

LHS = RHS

Thus,

Dividend = Divisor×Quotient + Remainder

Hence verified .

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Also ,

We can also verify the division rule using another factor of the given polynomial .

Now,

Dividend = Divisor×Quotient + Remainder

Thus,

=> x³ - 5x² + 8x - 4 = (x²- 4x + 4) × (x - 1) + 0

=> x³ - 5x² + 8x - 4 = (x² - 4x + 4) × (x - 1)

=> x³ - 5x² + 8x - 4 = (x - 2)²(x - 1)

=> x³ - 5x² + 8x - 4 = (x - 1)(x - 2)(x - 2)

Thus,

The given cubic polynomial can be rewritten as

x³ - 5x² + 8x - 4 = (x - 1)(x - 2)(x - 2) .

Hence,

The factors of the given cubic polynomial are ;

(x - 1) , (x - 2) , (x - 2)

Thus,

Let's use another factor (x - 2) to verify the division rule .

Thus,

Let's divide the given polynomial by (x-2) .

x - 2 ) x³ - 5x² + 8x - 4 ( x² - 3x + 2

x³ - 2x²

– +

- 3x² + 8x

- 3x² + 6x

+ -

2x - 4

2x - 4

– +

× ×

Here ,

Dividend = x³ - 5x² + 8x - 4

Divisor = x - 2

Quotient = x² - 3x + 2

Remainder = 0

Also,

We know that ;

Dividend = Divisor×Quotient + Remainder

Now,

=> LHS = Dividend

=> LHS = x³ - 5x² + 8x - 4

Also,

=> RHS = Divisor × Quotient + Remainder

=> RHS = (x² - 3x + 2) × (x - 2) + 0

=> RHS = x³ - 2x² - 3x² + 6x + 2x - 4

=> RHS = x³ - 5x² + 8x - 4

Clearly,

LHS = RHS

Thus,

Dividend = Divisor×Quotient + Remainder

Hence verified .