x^2+7x+14 divided by (x+3)
Answers
(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)
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)