(x^3+19x-30) Divide by
(x^2-3x-10)
Answers
Answer:
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "x2" was replaced by "x^2". 1 more similar replacement(s).
STEP
1
:
x3 - 19x - 30
Simplify —————————————
x2 - 3x - 10
Polynomial Roots Calculator :
1.1 Find roots (zeroes) of : F(x) = x3 - 19x - 30
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 -30.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,5 ,6 ,10 ,15 ,30
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -12.00
-2 1 -2.00 0.00 x + 2
-3 1 -3.00 0.00 x + 3
-5 1 -5.00 -60.00
-6 1 -6.00 -132.00
-10 1 -10.00 -840.00
-15 1 -15.00 -3120.00
-30 1 -30.00 -26460.00
1 1 1.00 -48.00
2 1 2.00 -60.00
3 1 3.00 -60.00
5 1 5.00 0.00 x - 5
6 1 6.00 72.00
10 1 10.00 780.00
15 1 15.00 3060.00
30 1 30.00 26400.00
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 - 19x - 30
can be divided by 3 different polynomials,including by x - 5
Polynomial Long Division :
1.2 Polynomial Long Division
Dividing : x3 - 19x - 30
("Dividend")
By : x - 5 ("Divisor")
dividend x3 - 19x - 30
- divisor * x2 x3 - 5x2
remainder 5x2 - 19x - 30
- divisor * 5x1 5x2 - 25x
remainder 6x - 30
- divisor * 6x0 6x - 30
remainder 0
Quotient : x2+5x+6 Remainder: 0
Trying to factor by splitting the middle term
1.3 Factoring x2+5x+6
The first term is, x2 its coefficient is 1 .
The middle term is, +5x its coefficient is 5 .
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 5 .
-6 + -1 = -7
-3 + -2 = -5
-2 + -3 = -5
-1 + -6 = -7
1 + 6 = 7
2 + 3 = 5 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)
Step-5 : Add up the four terms of step 4 :
(x+3) • (x+2)
Which is the desired factorization
Trying to factor by splitting the middle term
1.4 Factoring x2-3x-10
The first term is, x2 its coefficient is 1 .
The middle term is, -3x its coefficient is -3 .
The last term, "the constant", is -10
Step-1 : Multiply the coefficient of the first term by the constant 1 • -10 = -10
Step-2 : Find two factors of -10 whose sum equals the coefficient of the middle term, which is -3 .
-10 + 1 = -9
-5 + 2 = -3 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -5 and 2
x2 - 5x + 2x - 10
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-5)
Add up the last 2 terms, pulling out common factors :
2 • (x-5)
Step-5 : Add up the four terms of step 4 :
(x+2) • (x-5)
Which is the desired factorization
Canceling Out :
1.5 Cancel out (x+2) which appears on both sides of the fraction line.
Canceling Out :
1.6 Cancel out (x-5) which appears on both sides of the fraction line.
Final result :
x + 3
Step-by-step explanation:
Answer:
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