(e) 15x³ + 37x² + 53x + 55 by 3x + 5
please answer
Answers
Answer:
15x3 + 37x2 + 53x + 55
Simplify ——————————————————————
3x + 5
Checking for a perfect cube :
3.1 15x3 + 37x2 + 53x + 55 is not a perfect cube
Trying to factor by pulling out :
3.2 Factoring: 15x3 + 37x2 + 53x + 55
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 53x + 55
Group 2: 15x3 + 37x2
Pull out from each group separately :
Group 1: (53x + 55) • (1)
Group 2: (15x + 37) • (x2)
Step-by-step explanation:
3.3 Find roots (zeroes) of : F(x) = 15x3 + 37x2 + 53x + 55
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 15 and the Trailing Constant is 55.
The factor(s) are:
of the Leading Coefficient : 1,3 ,5 ,15
of the Trailing Constant : 1 ,5 ,11 ,55
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 24.00
-1 3 -0.33 40.89
-1 5 -0.20 45.76
-1 15 -0.07 51.63
-5 1 -5.00 -1160.00
-5 3 -1.67 0.00 3x + 5
-11 1 -11.00 -16016.00
-11 3 -3.67 -381.33
-11 5 -2.20 -42.24
-11 15 -0.73 30.12
-55 1 -55.00 -2386560.00
-55 3 -18.33 -80911.11
1 1 1.00 160.00
1 3 0.33 77.33
1 5 0.20 67.20
1 15 0.07 58.70
5 1 5.00 3120.00
5 3 1.67 315.56
11 1 11.00 25080.00
11 3 3.67 1486.22
11 5 2.20 510.40
11 15 0.73 119.68
55 1 55.00 2610520.00
55 3 18.33 105893.33
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
15x3 + 37x2 + 53x + 55
can be divided with 3x + 5
Polynomial Long Division :
3.4 Polynomial Long Division
Dividing : 15x3 + 37x2 + 53x + 55
("Dividend")
By : 3x + 5 ("Divisor")
dividend 15x3 + 37x2 + 53x + 55
- divisor * 5x2 15x3 + 25x2
remainder 12x2 + 53x + 55
- divisor * 4x1 12x2 + 20x
remainder 33x + 55
- divisor * 11x0 33x + 55
remainder 0
Quotient : 5x2+4x+11 Remainder: 0
Trying to factor by splitting the middle term
3.5 Factoring 5x2+4x+11
The first term is, 5x2 its coefficient is 5 .
The middle term is, +4x its coefficient is 4 .
The last term, "the constant", is +11
Step-1 : Multiply the coefficient of the first term by the constant 5 • 11 = 55
Step-2 : Find two factors of 55 whose sum equals the coefficient of the middle term, which is 4 .
-55 + -1 = -56
-11 + -5 = -16
-5 + -11 = -16
-1 + -55 = -56
1 + 55 = 56
5 + 11 = 16
11 + 5 = 16
55 + 1 = 56
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Canceling Out :
3.6 Cancel out (3x+5) which appears on both sides of the fraction line