4x^3-3x+9 is divided by 2x-3
Answers
STEP 1:
Equation at the end of step 1
STEp 2:
4x3 - 3x + 9
Simplify ————————————
2x - 3
2.1 Find roots (zeroes) of : F(x) = 4x3 - 3x + 9
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 4 and the Trailing Constant is 9.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4
of the Trailing Constant : 1 ,3 ,9
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 8.00
-1 2 -0.50 10.00
-1 4 -0.25 9.69
-3 1 -3.00 -90.00
-3 2 -1.50 0.00 2x + 3
Note - For tidiness, printing of 13 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
4x3 - 3x + 9
can be divided with 2x + 3
Polynomial Long Division :
2.2 Polynomial Long Division
Dividing : 4x3 - 3x + 9
("Dividend")
By : 2x + 3 ("Divisor")
dividend 4x3 - 3x + 9
- divisor * 2x2 4x3 + 6x2
remainder - 6x2 - 3x + 9
- divisor * -3x1 - 6x2 - 9x
remainder 6x + 9
- divisor * 3x0 6x + 9
remainder 0
Quotient : 2x2-3x+3 Remainder: 0
Trying to factor by splitting the middle term
2.3 Factoring 2x2-3x+3
The first term is, 2x2 its coefficient is 2 .
The middle term is, -3x its coefficient is -3 .
The last term, "the constant", is +3
Step-1 : Multiply the coefficient of the first term by the constant 2 • 3 = 6
Step-2 : Find two factors of 6 whose sum equals the coefficient of the middle term, which is -3 .
-6 + -1 = -7
-3 + -2 = -5
-2 + -3 = -5
-1 + -6 = -7
1 + 6 = 7
2 + 3 = 5
3 + 2 = 5
6 + 1 = 7
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
2.4 Polynomial Long Division
Dividing : 2x2-3x+3
("Dividend")
By : 2x-3 ("Divisor")
dividend 2x2 - 3x + 3
- divisor * x1 2x2 - 3x
remainder 3
- divisor * 0x0
remainder 3
Quotient : x
Remainder : 3
Final result :
(2x2 - 3x + 3) • (2x + 3)
———————————
2x - 3
Answer:
-2+5
Step-by-step explanation: