Question

Help: What is the quotient of 4x^3 + 5x^2 - 12 + 27/(x + 3)?

A. 4x^2 - 7x + 9
B. 4x^2 + 7x + 9
C. 4x^2 - 7x + 27
D. 4x^2 + 7x + 27

I think it's C.

Answer 1

umm hold on

Answer 2

@bohotness
she is really smart

Answer 3

Final result :

(3x2 - 2x + 1) • (x + 2) • -1
Step by step solution :

Step 1 :

Raise x to the 2nd power
Exponentiaion :

Equation at the end of step 1 :

((((((6•(x6))+(20•(x5)))+(7•(x4)))+(27•(x2)))-14x)+8)
—————————————————————————————————————————————————————
((((0-(2•(x3)))-(4•x2))+x)-4)
Step 2 :

Raise x to the 3rd power
Exponentiaion :

Equation at the end of step 2 :

((((((6•(x6))+(20•(x5)))+(7•(x4)))+(27•(x2)))-14x)+8)
—————————————————————————————————————————————————————
((((0-(2•x3))-22x2)+x)-4)
Step 3 :

Raise x to the 2nd power
Exponentiaion :

Equation at the end of step 3 :

((((((6•(x6))+(20•(x5)))+(7•(x4)))+(27•x2))-14x)+8)
———————————————————————————————————————————————————
(-2x3-4x2+x-4)
Step 4 :

Raise x to the 4th power
Exponentiaion :

Equation at the end of step 4 :

((((((6•(x6))+(20•(x5)))+(7•x4))+33x2)-14x)+8)
——————————————————————————————————————————————
(-2x3-4x2+x-4)
Step 5 :

Raise x to the 5th power
Exponentiaion :

Equation at the end of step 5 :

((((((6•(x6))+(20•x5))+7x4)+33x2)-14x)+8)
—————————————————————————————————————————
(-2x3-4x2+x-4)
Step 6 :

Raise x to the 6th power
Exponentiaion :

Equation at the end of step 6 :

((((((6•x6)+(22•5x5))+7x4)+33x2)-14x)+8)
————————————————————————————————————————
(-2x3-4x2+x-4)
Step 7 :

6x6 + 20x5 + 7x4 + 27x2 - 14x + 8
Simplify —————————————————————————————————
-2x3 - 4x2 + x - 4
Trying to factor by pulling out :

7.1 Factoring: 6x6 + 20x5 + 7x4 + 27x2 - 14x + 8

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

Group 1: 7x4 + 27x2
Group 2: 6x6 + 20x5
Group 3: -14x + 8

Pull out from each group separately :

Group 1: (7x2 + 27) • (x2)
Group 2: (3x + 10) • (2x5)
Group 3: (7x - 4) • (-2)


Looking for common sub-expressions :

Group 1: (7x2 + 27) • (x2)
Group 3: (7x - 4) • (-2)
Group 2: (3x + 10) • (2x5)

Bad news !! Factoring by pulling out fails :

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

Pulling out like terms :

7.2 Pull out like factors :

-2x3 - 4x2 + x - 4 =

-1 • (2x3 + 4x2 - x + 4)

Checking for a perfect cube :

7.3 2x3 + 4x2 - x + 4 is not a perfect cube

Trying to factor by pulling out :

7.4 Factoring: 2x3 + 4x2 - x + 4

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

Group 1: -x + 4
Group 2: 4x2 + 2x3

Pull out from each group separately :

Group 1: (-x + 4) • (1) = (x - 4) • (-1)
Group 2: (x + 2) • (2x2)

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 :

7.5 Find roots (zeroes) of : F(x) = 6x6 + 20x5 + 7x4 + 27x2 - 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 6 and the Trailing Constant is 8.

The factor(s) are:

of the Leading Coefficient : 1,2 ,3 ,6
of the Trailing Constant : 1 ,2 ,4 ,8

Let us test ....

P Q P/Q F(P/Q) Divisor
-1 1 -1.00 42.00
-1 2 -0.50 21.66
-1 3 -0.33 15.68
-1 6 -0.17 11.09
-2 1 -2.00 0.00 x + 2

Note - For tidiness, printing of 15 checks which found no root was suppressed

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
6x6 + 20x5 + 7x4 + 27x2 - 14x + 8
can be divided with x + 2

Polynomial Long Division :

7.6 Polynomial Long Division
Dividing : 6x6 + 20x5 + 7x4 + 27x2 - 14x + 8
("Dividend")
By : x + 2 ("Divisor")

dividend 6x6 + 20x5 + 7x4 + 27x2 - 14x + 8
- divisor * 6x5 6x6 + 12x5
remainder 8x5 + 7x4 + 27x2 - 14x + 8
- divisor * 8x4 8x5 + 16x4
remainder - 9x4 + 27x2 - 14x + 8
- divisor * -9x3 - 9x4 - 18x3
remainder 18x3 + 27x2 - 14x + 8
- divisor * 18x2 18x3 + 36x2
remainder - 9x2 - 14x + 8
- divisor * -9x1 - 9x2 - 18x
remainder 4x + 8
- divisor * 4x0 4x + 8
remainder 0
Quotient : 6x5+8x4-9x3+18x2-9x+4 Remainder: 0

Polynomial Roots Calculator :

7.7 Find roots (zeroes) of : F(x) = 6x5+8x4-9x3+18x2-9x+4

See theory in step 7.5
In this case, the Leading Coefficient is 6 and the Trailing Constant is 4.

The factor(s) are:

of the Leading Coefficient : 1,2 ,3 ,6
of the Trailing Constant : 1 ,2 ,4

Let us test ....

P Q P/Q F(P/Q) Divisor
-1 1 -1.00 42.00
-1 2 -0.50 14.44
-1 3 -0.33 9.41
-1 6 -0.17 6.05
-2 1 -2.00 102.00
-2 3 -0.67 21.46
-4 1 -4.00 -3192.00
-4 3 -1.33 69.33
1 1 1.00 18.00
1 2 0.50 3.56
1 3 0.33 2.79
1 6 0.17 2.97
2 1 2.00 306.00
2 3 0.67 5.70
4 1 4.00 7872.00
4 3 1.33 53.23

Polynomial Roots Calculator found no rational roots

Polynomial Roots Calculator :

7.8 Find roots (zeroes) of : F(x) = 2x3+4x2-x+4

See theory in step 7.5
In this case, the Leading Coefficient is 2 and the Trailing Constant is 4.

The factor(s) are:

of the Leading Coefficient : 1,2
of the Trailing Constant : 1 ,2 ,4

Let us test ....

P Q P/Q F(P/Q) Divisor
-1 1 -1.00 7.00
-1 2 -0.50 5.25
-2 1 -2.00 6.00
-4 1 -4.00 -56.00
1 1 1.00 9.00
1 2 0.50 4.75
2 1 2.00 34.00
4 1 4.00 192.00

Polynomial Roots Calculator found no rational roots

Polynomial Long Division :

7.9 Polynomial Long Division
Dividing : 6x5+8x4-9x3+18x2-9x+4
("Dividend")
By : 2x3+4x2-x+4 ("Divisor")

dividend 6x5 + 8x4 - 9x3 + 18x2 - 9x + 4
- divisor * 3x2 6x5 + 12x4 - 3x3 + 12x2
remainder - 4x4 - 6x3 + 6x2 - 9x + 4
- divisor * -2x1 - 4x4 - 8x3 + 2x2 - 8x
remainder 2x3 + 4x2 - x + 4
- divisor * x0 2x3 + 4x2 - x + 4
remainder 0
Quotient : 3x2-2x+1
Remainder : 0

Trying to factor by splitting the middle term

7.10 Factoring 3x2-2x+1

The first term is, 3x2 its coefficient is 3 .
The middle term is, -2x its coefficient is -2 .
The last term, "the constant", is +1

Step-1 : Multiply the coefficient of the first term by the constant 3 • 1 = 3

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

-3 + -1 = -4
-1 + -3 = -4
1 + 3 = 4
3 + 1 = 4

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Final result :

(3x2 - 2x + 1) • (x + 2) • -1