Question

X^2-3x-10/x+2

A. X-5; where x=-2
B. X-5; where x=5
C. 1/x-2
D. X-2

Answer 1

Step by step solution :
Step 1 :

10
Simplify x2-3x - ——
x

Rewriting the whole as an Equivalent Fraction :

1.1 Subtracting a fraction from a whole

Rewrite the whole as a fraction using x as the denominator :

x2 - 3x (x2 - 3x) • x
x2 - 3x = ——————— = —————————————
1 x

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator
Pulling out like terms :

1.2 Pull out like factors :

x2 - 3x = x • (x - 3)
Adding fractions that have a common denominator :

1.3 Adding up the two equivalent fractions

Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

x • (x-3) • x - (10) x3 - 3x2 - 10
———————————————————— = —————————————
x x

Equation at the end of step 1 :

(x3 - 3x2 - 10)
——————————————— + 2
x

Step 2 :

x3-3x2-10
Simplify ————————— + 2
x

Rewriting the whole as an Equivalent Fraction :

2.1 Adding a whole to a fraction

Rewrite the whole as a fraction using x as the denominator :

2 2 • x
2 = — = —————
1 x

Polynomial Roots Calculator :

2.2 Find roots (zeroes) of
F(x) = x3 - 3x2 - 10

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 -10.

The factor(s) are:

of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,5 ,10

Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -14.00
-2 1 -2.00 -30.00
-5 1 -5.00 -210.00
-10 1 -10.00 -1310.00
1 1 1.00 -12.00
2 1 2.00 -14.00
5 1 5.00 40.00
10 1 10.00 690.00


Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :

2.3 Adding up the two equivalent fractions

(x3-3x2-10) + 2 • x x3 - 3x2 + 2x - 10
——————————————————— = ——————————————————
x x

Checking for a perfect cube :

2.4 x3 - 3x2 + 2x - 10 is not a perfect cube
Trying to factor by pulling out :

2.5 Factoring: x3 - 3x2 + 2x - 10

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

Group 1: 2x - 10
Group 2: x3 - 3x2

Pull out from each group separately :

Group 1: (x - 5) • (2)
Group 2: (x - 3) • (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 :

2.6 Find roots (zeroes) of
F(x) = x3 - 3x2 + 2x - 10

See theory in step 2.2

In this case, the Leading Coefficient is 1 and the Trailing Constant is -10.

The factor(s) are:

of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,5 ,10

Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -16.00
-2 1 -2.00 -34.00
-5 1 -5.00 -220.00
-10 1 -10.00 -1330.00
1 1 1.00 -10.00
2 1 2.00 -10.00
5 1 5.00 50.00
10 1 10.00 710.00


Polynomial Roots Calculator found no rational roots
Final result :

x3 - 3x2 + 2x - 10
——————————————————
x