Step by step solution :
Step 1 :Equation at the over of step 1 : ((x4) - 3x2) - 4 = 0
Step 2 :Trying to lớn factor by splitting the middle term2.1Factoring 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 -4Step-1 : Multiply the coefficient of the first term by the constant 1•-4=-4Step-2 : Find two factors of -4 whose sum equals the coefficient of the middle term, which is -3.
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Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step2above, -4 and 1x4 - 4x2+1x2 - 4Step-4 : showroom up the first 2 terms, pulling out lượt thích factors:x2•(x2-4) địa chỉ up the last 2 terms, pulling out common factors:1•(x2-4) Step-5:Add up the four terms of step4:(x2+1)•(x2-4)Which is the desired factorization
Polynomial Roots Calculator :
2.2 Find roots (zeroes) of : F(x) = x2+1Polynomial Roots Calculator is a phối 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 và the Trailing Constant is 1. The factor(s) are: of the Leading Coefficient : 1of the Trailing Constant : 1 Let us chạy thử ....
Polynomial Roots Calculator found no rational roots
Trying lớn factor as a Difference of Squares:2.3 Factoring: x2-4 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 = bố is the commutative property of multiplication. Lưu ý : -AB+ AB equals zero and is therefore eliminated from the expression.Check: 4 is the square of 2Check: x2 is the square of x1Factorization is :(x + 2)•(x - 2)Equation at the kết thúc of step 2 :
(x2 + 1) • (x + 2) • (x - 2) = 0
Step 3 :Theory - Roots of a sản phẩm :3.1 A product 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 to lớn 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:3.2Solve: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. It has 2 imaginary, or complex solutions.x= 0.0000 + 1.0000 i x= 0.0000 - 1.0000 i
Solving a Single Variable Equation:3.3Solve:x+2 = 0Subtract 2 from both sides of the equation:x = -2
Solving a Single Variable Equation:3.4Solve:x-2 = 0Add 2 khổng lồ both sides of the equation:x = 2
Supplement : Solving Quadratic Equation DirectlySolving x4-3x2-4 = 0 directly Earlier we factored this polynomial by splitting the middle term.
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Let us now solve the equation by Completing The Square và by using the Quadratic Formula
Solving a Single Variable Equation:Equations which are reducible khổng lồ quadratic :
4.1Solvex4-3x2-4 = 0This equation is reducible to lớn quadratic. What this means is that using a new variable, we can rewrite this equation as a quadratic equation Using w, such that w = x2 transforms the equation into:w2-3w-4 = 0Solving this new equation using the quadratic formula we get two real solutions: 4.0000or-1.0000Now that we know the value(s) of w, we can calculate x since x is √ w Doing just this we discover that the solutions of x4-3x2-4 = 0are either:x =√ 4.000 = 2.00000 or:x =√ 4.000 = -2.00000 or:x =√-1.000 = 0.0 + 1.00000 i or:x =√-1.000 = 0.0 - 1.00000 i