x^6-x^4-x^2+1=0
This solution giao dịch with factoring binomials using the difference of squares.
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Step by Step Solution
Step by step solution :
Step 1 :
Checking for a perfect cube :1.1x6-x4-x2+1 is not a perfect cube Trying to lớn factor by pulling out :1.2 Factoring: x6-x4-x2+1 Thoughtfully split the expression at hand into groups, each group having two terms:Group 1: -x2+1Group 2: x6-x4Pull out from each group separately :Group 1: (-x2+1) • (1) = (x2-1) • (-1)Group 2: (x2-1) • (x4) -------------------Add up the two groups:(x2-1) • (x4-1)Which is the desired factorization
Trying to factor as a Difference of Squares:1.3 Factoring: x4-1 Theory : A difference of two perfect squares, A2-B2can be factored into (A+B)•(A-B)Proof:(A+B)•(A-B)= A2 - AB+BA-B2= A2 -AB+ AB - B2 = A2 - B2Note : AB = tía is the commutative property of multiplication. Cảnh báo : -AB+ AB equals zero and is therefore eliminated from the expression.Check: 1 is the square of 1Check: x4 is the square of x2Factorization is :(x2 + 1)•(x2 - 1)
Polynomial Roots Calculator :
1.4 Find roots (zeroes) of : F(x) = x2 + 1Polynomial Roots Calculator is a mix of methods aimed at finding values ofxfor which F(x)=0 Rational Roots demo 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 integersThe Rational Root Theorem states that if a polynomial zeroes for a rational numberP/Q then p is a factor of the Trailing Constant and Q is a factor of the Leading CoefficientIn this case, the Leading Coefficient is 1 and the Trailing Constant is 1. The factor(s) are: of the Leading Coefficient : 1of the Trailing Constant : 1 Let us demo ....
| -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:1.5 Factoring: x2 - 1 Check: 1 is the square of 1Check: x2 is the square of x1Factorization is :(x + 1)•(x - 1)
Trying khổng lồ factor as a Difference of Squares:1.6 Factoring: x2 - 1 Check: 1 is the square of 1Check: x2 is the square of x1Factorization is :(x + 1)•(x - 1)
Multiplying Exponential Expressions:1.7 Multiply (x + 1) by (x + 1)The rule says : lớn multiply exponential expressions which have the same base, địa chỉ cửa hàng up their exponents.In our case, the common base is (x+1) và the exponents are:1,as(x+1) is the same number as (x+1)1and1,as(x+1) is the same number as (x+1)1The sản phẩm is therefore, (x+1)(1+1) = (x+1)2
Multiplying Exponential Expressions:1.8 Multiply (x-1) by (x-1)The rule says : to lớn multiply exponential expressions which have the same base, add up their exponents.In our case, the common base is (x-1) và the exponents are:1,as(x-1) is the same number as (x-1)1and1,as(x-1) is the same number as (x-1)1The sản phẩm is therefore, (x-1)(1+1) = (x-1)2
Equation at the kết thúc of step 1 :(x2 + 1) • (x + 1)2 • (x - 1)2 = 0
Step 2 :
Theory - Roots of a sản phẩm :2.1 A hàng hóa of several terms equals zero.When a product of two or more terms equals zero, then at least one of the terms must be zero.We shall now solve each term = 0 separatelyIn other words, we are going khổng lồ solve as many equations as there are terms in the productAny solution of term = 0 solves product = 0 as well.Solving a Single Variable Equation:2.2Solve:x2+1 = 0Subtract 1 from both sides of the equation:x2 = -1 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get: x = ± √ -1 In Math,iis called the imaginary unit. It satisfies i2=-1. Both i & -i are the square roots of -1The equation has no real solutions.
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It has 2 imaginary, or complex solutions.x= 0.0000 + 1.0000 i x= 0.0000 - 1.0000 i
Solving a Single Variable Equation:2.3Solve:(x+1)2 = 0(x+1)2 represents, in effect, a sản phẩm of 2 terms which is equal to zero For the sản phẩm to be zero, at least one of these terms must be zero. Since all these terms are equal to each other, it actually means: x+1=0 Subtract 1 from both sides of the equation:x = -1
Solving a Single Variable Equation:2.4Solve:(x-1)2 = 0As explained in step #02.03 above, the equation to lớn be solved isx-1 = 0Add 1 to lớn both sides of the equation:x = 1