(ii) 2x4 -5x+3x3-3+3x2 by 2x2-x-1 divide
Answers
Answer:
(2x + 3) • (x - 1)
Step-by-step explanation:
(1): "x2" was replaced by "x^2". 3 more similar replacement(s).
STEP
1
:
Equation at the end of step 1
STEP
2
:
Equation at the end of step
2
:
STEP
3
:
Equation at the end of step
3
:
STEP
4
:
2x4 - 3x3 - 3x2 + 7x - 3
Simplify ————————————————————————
x2 - 2x + 1
Polynomial Roots Calculator :
4.1 Find roots (zeroes) of : F(x) = 2x4 - 3x3 - 3x2 + 7x - 3
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 2 and the Trailing Constant is -3.
The factor(s) are:
of the Leading Coefficient : 1,2
of the Trailing Constant : 1 ,3
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -8.00
-1 2 -0.50 -6.75
-3 1 -3.00 192.00
-3 2 -1.50 0.00 2x + 3
1 1 1.00 0.00 x - 1
1 2 0.50 -0.50
3 1 3.00 72.00
3 2 1.50 0.75
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
2x4 - 3x3 - 3x2 + 7x - 3
can be divided by 2 different polynomials,including by x - 1
Polynomial Long Division :
4.2 Polynomial Long Division
Dividing : 2x4 - 3x3 - 3x2 + 7x - 3
("Dividend")
By : x - 1 ("Divisor")
dividend 2x4 - 3x3 - 3x2 + 7x - 3
- divisor * 2x3 2x4 - 2x3
remainder - x3 - 3x2 + 7x - 3
- divisor * -x2 - x3 + x2
remainder - 4x2 + 7x - 3
- divisor * -4x1 - 4x2 + 4x
remainder 3x - 3
- divisor * 3x0 3x - 3
remainder 0
Quotient : 2x3-x2-4x+3 Remainder: 0
Polynomial Roots Calculator :
4.3 Find roots (zeroes) of : F(x) = 2x3-x2-4x+3
See theory in step 4.1
In this case, the Leading Coefficient is 2 and the Trailing Constant is 3.
The factor(s) are:
of the Leading Coefficient : 1,2
of the Trailing Constant : 1 ,3
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 4.00
-1 2 -0.50 4.50
-3 1 -3.00 -48.00
-3 2 -1.50 0.00 2x+3
1 1 1.00 0.00 x-1
1 2 0.50 1.00
3 1 3.00 36.00
3 2 1.50 1.50
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
2x3-x2-4x+3
can be divided by 2 different polynomials,including by x-1
Polynomial Long Division :
4.4 Polynomial Long Division
Dividing : 2x3-x2-4x+3
("Dividend")
By : x-1 ("Divisor")
dividend 2x3 - x2 - 4x + 3
- divisor * 2x2 2x3 - 2x2
remainder x2 - 4x + 3
- divisor * x1 x2 - x
remainder - 3x + 3
- divisor * -3x0 - 3x + 3
remainder 0
Quotient : 2x2+x-3 Remainder: 0
Trying to factor by splitting the middle term
4.5 Factoring 2x2+x-3
The first term is, 2x2 its coefficient is 2 .
The middle term is, +x its coefficient is 1 .
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 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
2x2 - 2x + 3x - 3
Step-4 : Add up the first 2 terms, pulling out like factors :
2x • (x-1)
Add up the last 2 terms, pulling out common factors :
3 • (x-1)
Step-5 : Add up the four terms of step 4 :
(2x+3) • (x-1)
Which is the desired factorization
Trying to factor by splitting the middle term
4.6 Factoring x2-2x+1
The first term is, x2 its coefficient is 1 .
The middle term is, -2x its coefficient is -2 .
The last term, "the constant", is +1
Step-1 : Multiply the coefficient of the first term by the constant 1 • 1 = 1
Step-2 : Find two factors of 1 whose sum equals the coefficient of the middle term, which is -2 .
-1 + -1 = -2 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -1 and -1
x2 - 1x - 1x - 1
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 :
1 • (x-1)
Step-5 : Add up the four terms of step 4 :
(x-1) • (x-1)
Which is the desired factorization
Canceling Out :
4.7 Cancel out (x-1) which appears on both sides of the fraction line.
Canceling Out :
4.8 Cancel out (x-1) which appears on both sides of the fraction line.
Final result :
(2x + 3) • (x - 1)
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the answer is 91
Step-by-step explanation:
i hope it can help u