This solution deals with finding the roots (zeroes) of polynomials.
Bạn đang xem: Calculus
Step by Step Solution
Step by step solution :
Step 1 :Equation at the over of step 1 : (((4 • (x3)) - 2x2) - 6x) - 2 = 0
Step 2 :Equation at the over of step 2 : ((22x3 - 2x2) - 6x) - 2 = 0
Step 3 :
Step 4 :Pulling out like terms :4.1 Pull out lượt thích factors:4x3 - 2x2 - 6x - 2=2•(2x3 - x2 - 3x - 1)Checking for a perfect cube :
4.22x3 - x2 - 3x - 1 is not a perfect cubeTrying to lớn factor by pulling out :
4.3 Factoring: 2x3 - x2 - 3x - 1 Thoughtfully split the expression at hand into groups, each group having two terms:Group 1: -3x - 1Group 2: 2x3 - x2Pull out from each group separately :Group 1: (3x + 1) • (-1)Group 2: (2x - 1) • (x2)Bad news !! Factoring by pulling out fails : The groups have no common factor và can not be added up to form a multiplication.
Polynomial Roots Calculator :
4.4 Find roots (zeroes) of : F(x) = 2x3 - x2 - 3x - 1Polynomial Roots Calculator is a set of methods aimed at finding values ofxfor which F(x)=0 Rational Roots chạy thử 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 2 & the Trailing Constant is -1. The factor(s) are: of the Leading Coefficient : 1,2 of the Trailing Constant : 1 Let us chạy thử ....
|-1||2||-0.50||0.00||2x + 1|
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p chú ý that q and p originate from P/Q reduced khổng lồ its lowest terms In our case this means that 2x3 - x2 - 3x - 1can be divided with 2x + 1
Polynomial Long Division :
4.5 Polynomial Long Division Dividing : 2x3 - x2 - 3x - 1("Dividend") By:2x + 1("Divisor")
Quotient : x2-x-1 Remainder: 0Trying to lớn factor by splitting the middle term
4.6Factoring x2-x-1 The first term is, x2 its coefficient is 1.The middle term is, -x its coefficient is -1.The last term, "the constant", is -1Step-1 : Multiply the coefficient of the first term by the constant 1•-1=-1Step-2 : Find two factors of -1 whose sum equals the coefficient of the middle term, which is -1.
Observation : No two such factors can be found !! Conclusion : Trinomial can not be factoredEquation at the over of step 4 :
2 • (x2 - x - 1) • (2x + 1) = 0
Step 5 :Theory - Roots of a product :5.1 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 to lớn solve as many equations as there are terms in the productAny solution of term = 0 solves hàng hóa = 0 as well.
Xem thêm: Karik Là Ai? Tiểu Sử Karik Là Ai? Chi Tiết Tiểu Sử Và Sự Nghiệp Của Ca Sĩ Karik
Equations which are never true:5.2Solve:2=0This equation has no solution. A a non-zero constant never equals zero.
Parabola, Finding the Vertex:5.3Find the Vertex ofy = x2-x-1Parabolas have a highest or a lowest point called the Vertex.Our parabola opens up và accordingly has a lowest point (AKA absolute minimum).We know this even before plotting "y" because the coefficient of the first term,1, is positive (greater than zero).Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x-intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.Parabolas can mã sản phẩm many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want khổng lồ be able lớn find the coordinates of the vertex.For any parabola,Ax2+Bx+C,the x-coordinate of the vertex is given by -B/(2A). In our case the x coordinate is 0.5000Plugging into the parabola formula 0.5000 for x we can calculate the y-coordinate:y = 1.0 * 0.50 * 0.50 - 1.0 * 0.50 - 1.0 or y = -1.250Parabola, Graphing Vertex và X-Intercepts :
Root plot for : y = x2-x-1 Axis of Symmetry (dashed) x= 0.50 Vertex at x,y = 0.50,-1.25 x-Intercepts (Roots) : Root 1 at x,y = -0.62, 0.00 Root 2 at x,y = 1.62, 0.00Solve Quadratic Equation by Completing The Square
5.4Solvingx2-x-1 = 0 by Completing The Square.Add 1 khổng lồ both side of the equation : x2-x = 1Now the clever bit: Take the coefficient of x, which is 1, divide by two, giving 1/2, & finally square it giving 1/4Add 1/4 to both sides of the equation :On the right hand side we have:1+1/4or, (1/1)+(1/4)The common denominator of the two fractions is 4Adding (4/4)+(1/4) gives 5/4So adding to lớn both sides we finally get:x2-x+(1/4) = 5/4Adding 1/4 has completed the left hand side into a perfect square :x2-x+(1/4)=(x-(1/2))•(x-(1/2))=(x-(1/2))2 Things which are equal to lớn the same thing are also equal khổng lồ one another. Sincex2-x+(1/4) = 5/4 andx2-x+(1/4) = (x-(1/2))2 then, according to lớn the law of transitivity,(x-(1/2))2 = 5/4We"ll refer lớn this Equation as Eq. #5.4.1 The Square Root Principle says that When two things are equal, their square roots are equal.Note that the square root of(x-(1/2))2 is(x-(1/2))2/2=(x-(1/2))1=x-(1/2)Now, applying the Square Root Principle lớn Eq.#5.4.1 we get:x-(1/2)= √ 5/4 Add 50% to both sides to lớn obtain:x = 1/2 + √ 5/4 Since a square root has two values, one positive và the other negativex2 - x - 1 = 0has two solutions:x = 1/2 + √ 5/4 orx = một nửa - √ 5/4 note that √ 5/4 can be written as√5 / √4which is √5 / 2
Solve Quadratic Equation using the Quadratic Formula
5.5Solvingx2-x-1 = 0 by the Quadratic Formula.According khổng lồ the Quadratic Formula,x, the solution forAx2+Bx+C= 0 , where A, B and C are numbers, often called coefficients, is given by :-B± √B2-4ACx = ————————2A In our case,A= 1B= -1C= -1 Accordingly,B2-4AC=1 - (-4) = 5Applying the quadratic formula : 1 ± √ 5 x=————2 √ 5 , rounded to 4 decimal digits, is 2.2361So now we are looking at:x=(1± 2.236 )/2Two real solutions:x =(1+√5)/2= 1.618 or:x =(1-√5)/2=-0.618
Solving a Single Variable Equation:5.6Solve:2x+1 = 0Subtract 1 from both sides of the equation:2x = -1 Divide both sides of the equation by 2:x = -1/2 = -0.500