Math, asked by anamika183, 1 year ago

x^2+7x+14 divided by (x+3)​

Answers

Answered by Tasu06
4

(x3+7x2+14x+8)/(x+2)

Final result :

(x + 1) • (x + 4)

Step by step solution :

Step 1 :

Equation at the end of step 1 :

Step 2 :

x3 + 7x2 + 14x + 8

Simplify ——————————————————

x + 2

Checking for a perfect cube :

2.1 x3 + 7x2 + 14x + 8 is not a perfect cube

Trying to factor by pulling out :

2.2 Factoring: x3 + 7x2 + 14x + 8

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

Group 1: 14x + 8

Group 2: 7x2 + x3

Pull out from each group separately :

Group 1: (7x + 4) • (2)

Group 2: (x + 7) • (x2)

Bad news !! Factoring by pulling out fails :

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

Polynomial Roots Calculator :

2.3 Find roots (zeroes) of : F(x) = x3 + 7x2 + 14x + 8

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

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 1 and the Trailing Constant is 8.

The factor(s) are:

of the Leading Coefficient : 1

of the Trailing Constant : 1 ,2 ,4 ,8

Let us test ....

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

-1 1 -1.00 0.00 x + 1

-2 1 -2.00 0.00 x + 2

-4 1 -4.00 0.00 x + 4

-8 1 -8.00 -168.00

1 1 1.00 30.00

2 1 2.00 72.00

4 1 4.00 240.00

8 1 8.00 1080.00

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

x3 + 7x2 + 14x + 8

can be divided by 3 different polynomials,including by x + 4

Polynomial Long Division :

2.4 Polynomial Long Division

Dividing : x3 + 7x2 + 14x + 8

("Dividend")

By : x + 4 ("Divisor")

dividend x3 + 7x2 + 14x + 8

- divisor * x2 x3 + 4x2

remainder 3x2 + 14x + 8

- divisor * 3x1 3x2 + 12x

remainder 2x + 8

- divisor * 2x0 2x + 8

remainder 0

Quotient : x2+3x+2 Remainder: 0

Trying to factor by splitting the middle term

2.5 Factoring x2+3x+2

The first term is, x2 its coefficient is 1 .

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

The last term, "the constant", is +2

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

x2 + 1x + 2x + 2

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

x • (x+1)

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

2 • (x+1)

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

(x+2) • (x+1)

Which is the desired factorization

Canceling Out :

2.6 Cancel out (x+2) which appears on both sides of the fraction line.

Final result :

(x + 1) • (x + 4)

Answered by brunoconti
3

Answer:

Step-by-step explanation:

x^2 + 7x + 14 = (x + 3)(x + 4) + 2

(x^2 + 7x + 14)/(x + 3) = x + 4 + 2/(x + 3)

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