evaluate x⁴ + 3x² + 4
Answers
Step-by-step explanation:
Changes made to your input should not affect the solution:
(1): "x4" was replaced by "x^4".
STEP
1
:
Equation at the end of step 1
((x4) - 3x2) - 4
STEP
2
:
Trying to factor by splitting the middle term
2.1 Factoring x4-3x2-4
The first term is, x4 its coefficient is 1 .
The middle term is, -3x2 its coefficient is -3 .
The last term, "the constant", is -4
Step-1 : Multiply the coefficient of the first term by the constant 1 • -4 = -4
Step-2 : Find two factors of -4 whose sum equals the coefficient of the middle term, which is -3 .
-4 + 1 = -3 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -4 and 1
x4 - 4x2 + 1x2 - 4
Step-4 : Add up the first 2 terms, pulling out like factors :
x2 • (x2-4)
Add up the last 2 terms, pulling out common factors :
1 • (x2-4)
Step-5 : Add up the four terms of step 4 :
(x2+1) • (x2-4)
Which is the desired factorization
Polynomial Roots Calculator :
2.2 Find roots (zeroes) of : F(x) = x2+1
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 1.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 2.00
1 1 1.00 2.00
Polynomial Roots Calculator found no rational roots
Trying to factor as a Difference of Squares:
2.3 Factoring: x2-4
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : 4 is the square of 2
Check : x2 is the square of x1
Factorization is : (x + 2) • (x - 2)
Final result :
(x2 + 1) • (x + 2) • (x - 2)
Answer:
How can I factorize x^4-3x^2+2?
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Awnon Bhowmik
Answered 4 years ago
Very easy, put x=1 , the polynomial will evaluate to 0 . This means that (x−1) is a factor of the polynomial.
x4–3x2+2
=x4−x3+x3−x2–2x2+2
=x3(x−1)+x2(x−1)−2(x2–1)
=x3(x−1)+x2(x−1)−2(x+1)(x−1)
=(x−1){x3+x2–2(x+1)}
=(x−1)(x3+x2–2x−2)
=(x−1){x2(x+1)−2(x+1)}
=(x−1)(x+1)(x2−2)
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