Math, asked by chhavisingh78, 1 month ago

2x^3-x^2-5x+2 factorise by division method​

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Answered by Anonymous
3

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Factoring: 2x3-x2-5x-2

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1: -5x-2

Group 2: 2x3-x2

Pull out from each group separately :

Group 1: (5x+2) • (-1)

Group 2: (2x-1) • (x2)

Bad news !! Factoring by pulling out fails :

The groups have no common factor and can not be added up to form a multiplication.

The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient

In this case, the Leading Coefficient is 2 and the Trailing Constant is -2.

The factor(s) are:

of the Leading Coefficient : 1,2

of the Trailing Constant : 1 ,2

Let us test ....

P Q P/Q F(P/Q) Divisor

-1 1 -1.00 0.00 x+1

-1 2 -0.50 0.00 2x+1

-2 1 -2.00 -12.00

1 1 1.00 -6.00

1 2 0.50 -4.50

2 1 2.00 0.00 x-2

The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms

In our case this means that

2x3-x2-5x-2

can be divided by 3 different polynomials,including by x-2

Find roots (zeroes) of : F(x) = 2x3-x2-5x-2

Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers

Polynomial Long Division

Dividing : 2x3-x2-5x-2

("Dividend")

By : x-2 ("Divisor")

dividend 2x3 - x2 - 5x - 2

- divisor * 2x2 2x3 - 4x2

remainder 3x2 - 5x - 2

- divisor * 3x1 3x2 - 6x

remainder x - 2

- divisor * x0 x - 2

remainder 0

Quotient : 2x2+3x+1 Remainder: 0

Factoring 2x2+3x+1

The first term is, 2x2 its coefficient is 2 .

The middle term is, +3x its coefficient is 3 .

The last term, "the constant", is +1

Step-1 : Multiply the coefficient of the first term by the constant 2 • 1 = 2

Step-2 : Find two factors of 2 whose sum equals the coefficient of the middle term, which is 3 .

-2 + -1 = -3

-1 + -2 = -3

1 + 2 = 3 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 1 and 2

2x2 + 1x + 2x + 1

Step-4 : Add up the first 2 terms, pulling out like factors :

x • (2x+1)

Add up the last 2 terms, pulling out common factors :

1 • (2x+1)

Step-5 : Add up the four terms of step 4 :

(x+1) • (2x+1)

Which is the desired factorization

Final result :

(2x + 1) • (x + 1) • (x - 2)

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