Question

Factor x^3-3x^2-10x+24
Please show work thx

Answer 1

\[f(x)=x^3-3x^2-10x+24\]\[f(2)=8-12-20+24=0\]so (x-2) is a factor

Answer 2

\[(x^3-3x^2-10x+24) \div (x-2)=?\] then factorise as you would for a cubic expression.

Answer 3

Wow thx Dominicng!

Answer 4

Welcomed.

Answer 5

x3-3x2-10x+24

Final result :

(x + 3) • (x - 2) • (x - 4)
Step by step solution :

Step 1 :

Simplify x3-3x2-10x + 24
Checking for a perfect cube :

1.1 x3-3x2-10x+24 is not a perfect cube

Trying to factor by pulling out :

1.2 Factoring: x3-3x2-10x+24

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

Group 1: -10x+24
Group 2: x3-3x2

Pull out from each group separately :

Group 1: (5x-12) • (-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 :

1.3 Find roots (zeroes) of
F(x) = x3-3x2-10x+24
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 24.

The factor(s) are:

of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,8 ,12 ,24

Let us test ....

P Q P/Q F(P/Q) Divisor
-1 1 -1.00 30.00
-2 1 -2.00 24.00
-3 1 -3.00 0.00 x+3
-4 1 -4.00 -48.00
-6 1 -6.00 -240.00
-8 1 -8.00 -600.00
-12 1 -12.00 -2016.00
-24 1 -24.00 -15288.00
1 1 1.00 12.00
2 1 2.00 0.00 x-2
3 1 3.00 -6.00
4 1 4.00 0.00 x-4
6 1 6.00 72.00
8 1 8.00 264.00
12 1 12.00 1200.00
24 1 24.00 11880.00

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
x3-3x2-10x+24
can be divided by 3 different polynomials,including by x-4

Polynomial Long Division :

1.4 Polynomial Long Division
Dividing : x3-3x2-10x+24
("Dividend")
By : x-4 ("Divisor")
dividend x3 - 3x2 - 10x + 24
- divisor * x2 x3 - 4x2
remainder x2 - 10x + 24
- divisor * x1 x2 - 4x
remainder - 6x + 24
- divisor * -6x0 - 6x + 24
remainder 0
Quotient : x2+x-6
Remainder: 0

Trying to factor by splitting the middle term

1.5 Factoring x2+x-6

The first term is, x2 its coefficient is 1 .
The middle term is, +x its coefficient is 1 .
The last term, "the constant", is -6

Step-1 : Multiply the coefficient of the first term by the constant 1 • -6 = -6

Step-2 : Find two factors of -6 whose sum equals the coefficient of the middle term, which is 1 .

-6 + 1 = -5
-3 + 2 = -1
-2 + 3 = 1 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -2 and 3
x2 - 2x + 3x - 6

Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-2)
Add up the last 2 terms, pulling out common factors :
3 • (x-2)
Now add up the four terms of step 3 :
(x+3) • (x-2)
Which is the desired factorization

Final result :

(x + 3) • (x - 2) • (x - 4)