### 3x^4-2x^3-25x^2+28x+12=0

This solution đơn hàng with polynomial long division.

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## Step by Step Solution

## Step by step solution :

## Step 1 :

Equation at the kết thúc of step 1 : ((((3•(x4))-(2•(x3)))-52x2)+28x)+12 = 0## Step 2 :

Equation at the kết thúc of step 2 : ((((3•(x4))-2x3)-52x2)+28x)+12 = 0## Step 3 :

Equation at the kết thúc of step 3 : (((3x4 - 2x3) - 52x2) + 28x) + 12 = 0## Step 4 :

### Polynomial Roots Calculator :

4.1 Find roots (zeroes) of : F(x) = 3x4-2x3-25x2+28x+12Polynomial Roots Calculator is a phối of methods aimed at finding values ofxfor which F(x)=0 Rational Roots thử nghiệm 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 & Q is a factor of the Leading CoefficientIn this case, the Leading Coefficient is 3 & the Trailing Constant is 12. The factor(s) are: of the Leading Coefficient : 1,3 of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,12 Let us kiểm tra ....PQP/QF(P/Q)Divisor-1 | 1 | -1.00 | -36.00 | ||||||

-1 | 3 | -0.33 | 0.00 | 3x+1 | |||||

-2 | 1 | -2.00 | -80.00 | ||||||

-2 | 3 | -0.67 | -16.59 | ||||||

-3 | 1 | -3.00 | 0.00 | x+3 | |||||

-4 | 1 | -4.00 | 396.00 | ||||||

-4 | 3 | -1.33 | -55.56 | ||||||

-6 | 1 | -6.00 | 3264.00 | ||||||

-12 | 1 | -12.00 | 61740.00 | ||||||

1 | 1 | 1.00 | 16.00 | ||||||

1 | 3 | 0.33 | 18.52 | ||||||

2 | 1 | 2.00 | 0.00 | x-2 | |||||

2 | 3 | 0.67 | 19.56 | ||||||

3 | 1 | 3.00 | 60.00 | ||||||

4 | 1 | 4.00 | 364.00 | ||||||

4 | 3 | 1.33 | 9.63 | ||||||

6 | 1 | 6.00 | 2736.00 | ||||||

12 | 1 | 12.00 | 55500.00 |

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 3x4-2x3-25x2+28x+12can be divided by 3 different polynomials,including by x-2

### Polynomial Long Division :

4.2 Polynomial Long Division Dividing : 3x4-2x3-25x2+28x+12("Dividend") By:x-2("Divisor")

dividend | 3x4 | - | 2x3 | - | 25x2 | + | 28x | + | 12 | ||

-divisor | * 3x3 | 3x4 | - | 6x3 | |||||||

remainder | 4x3 | - | 25x2 | + | 28x | + | 12 | ||||

-divisor | * 4x2 | 4x3 | - | 8x2 | |||||||

remainder | - | 17x2 | + | 28x | + | 12 | |||||

-divisor | * -17x1 | - | 17x2 | + | 34x | ||||||

remainder | - | 6x | + | 12 | |||||||

-divisor | * -6x0 | - | 6x | + | 12 | ||||||

remainder | 0 |

Quotient : 3x3+4x2-17x-6 Remainder: 0

### Polynomial Roots Calculator :

4.3 Find roots (zeroes) of : F(x) = 3x3+4x2-17x-6See theory in step 4.1 In this case, the Leading Coefficient is 3 and the Trailing Constant is -6. The factor(s) are: of the Leading Coefficient : 1,3 of the Trailing Constant : 1 ,2 ,3 ,6 Let us thử nghiệm ....

PQP/QF(P/Q)Divisor-1 | 1 | -1.00 | 12.00 | ||||||

-1 | 3 | -0.33 | 0.00 | 3x+1 | |||||

-2 | 1 | -2.00 | 20.00 | ||||||

-2 | 3 | -0.67 | 6.22 | ||||||

-3 | 1 | -3.00 | 0.00 | x+3 | |||||

-6 | 1 | -6.00 | -408.00 | ||||||

1 | 1 | 1.00 | -16.00 | ||||||

1 | 3 | 0.33 | -11.11 | ||||||

2 | 1 | 2.00 | 0.00 | x-2 | |||||

2 | 3 | 0.67 | -14.67 | ||||||

3 | 1 | 3.00 | 60.00 | ||||||

6 | 1 | 6.00 | 684.00 |

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 lớn its lowest terms In our case this means that 3x3+4x2-17x-6can be divided by 3 different polynomials,including by x-2

### Polynomial Long Division :

4.4 Polynomial Long Division Dividing : 3x3+4x2-17x-6("Dividend") By:x-2("Divisor")

dividend | 3x3 | + | 4x2 | - | 17x | - | 6 | ||

-divisor | * 3x2 | 3x3 | - | 6x2 | |||||

remainder | 10x2 | - | 17x | - | 6 | ||||

-divisor | * 10x1 | 10x2 | - | 20x | |||||

remainder | 3x | - | 6 | ||||||

-divisor | * 3x0 | 3x | - | 6 | |||||

remainder | 0 |

Quotient : 3x2+10x+3 Remainder: 0

Trying lớn factor by splitting the middle term4.5Factoring 3x2+10x+3 The first term is, 3x2 its coefficient is 3.The middle term is, +10x its coefficient is 10.The last term, "the constant", is +3Step-1 : Multiply the coefficient of the first term by the constant 3•3=9Step-2 : Find two factors of 9 whose sum equals the coefficient of the middle term, which is 10.

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-9 | + | -1 | = | -10 | ||

-3 | + | -3 | = | -6 | ||

-1 | + | -9 | = | -10 | ||

1 | + | 9 | = | 10 | That"s it |

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step2above, 1 and 93x2 + 1x+9x + 3Step-4 : địa chỉ up the first 2 terms, pulling out like factors:x•(3x+1) địa chỉ cửa hàng up the last 2 terms, pulling out common factors:3•(3x+1) Step-5:Add up the four terms of step4:(x+3)•(3x+1)Which is the desired factorization

Multiplying Exponential Expressions:4.6 Multiply (x-2) by (x-2)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-2) & the exponents are:1,as(x-2) is the same number as (x-2)1and1,as(x-2) is the same number as (x-2)1The sản phẩm is therefore, (x-2)(1+1) = (x-2)2

Equation at the end of step 4 :(3x + 1) • (x + 3) • (x - 2)2 = 0

## Step 5 :

Theory - Roots of a sản phẩm :5.1 A product of several terms equals zero.When a sản phẩm 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 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:5.2Solve:3x+1 = 0Subtract 1 from both sides of the equation:3x = -1 Divide both sides of the equation by 3:x = -1/3 = -0.333

Solving a Single Variable Equation:5.3Solve:x+3 = 0Subtract 3 from both sides of the equation:x = -3

Solving a Single Variable Equation:5.4Solve:(x-2)2 = 0(x-2)2 represents, in effect, a hàng hóa of 2 terms which is equal to zero For the hàng hóa 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-2=0 showroom 2 khổng lồ both sides of the equation:x = 2

### Supplement : Solving Quadratic Equation Directly

Solving 3x2+10x+3 = 0 directly Earlier we factored this polynomial by splitting the middle term. Let us now solve the equation by Completing The Square and by using the Quadratic FormulaParabola, Finding the Vertex:6.1Find the Vertex ofy = 3x2+10x+3Parabolas have a highest or a lowest point called the Vertex.Our parabola opens up and accordingly has a lowest point (AKA absolute minimum).We know this even before plotting "y" because the coefficient of the first term,3, 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 model 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 lớn 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 -1.6667Plugging into the parabola formula -1.6667 for x we can calculate the y-coordinate:y = 3.0 * -1.67 * -1.67 + 10.0 * -1.67 + 3.0 or y = -5.333

Parabola, Graphing Vertex and X-Intercepts :Root plot for : y = 3x2+10x+3 Axis of Symmetry (dashed) x=-1.67 Vertex at x,y = -1.67,-5.33 x-Intercepts (Roots) : Root 1 at x,y = -3.00, 0.00 Root 2 at x,y = -0.33, 0.00

Solve Quadratic Equation by Completing The Square6.2Solving3x2+10x+3 = 0 by Completing The Square.Divide both sides of the equation by 3 to have 1 as the coefficient of the first term :x2+(10/3)x+1 = 0Subtract 1 from both side of the equation :x2+(10/3)x = -1Now the clever bit: Take the coefficient of x, which is 10/3, divide by two, giving 5/3, và finally square it giving 25/9Add 25/9 to lớn both sides of the equation :On the right hand side we have:-1+25/9or, (-1/1)+(25/9)The common denominator of the two fractions is 9Adding (-9/9)+(25/9) gives 16/9So adding khổng lồ both sides we finally get:x2+(10/3)x+(25/9) = 16/9Adding 25/9 has completed the left hand side into a perfect square :x2+(10/3)x+(25/9)=(x+(5/3))•(x+(5/3))=(x+(5/3))2 Things which are equal khổng lồ the same thing are also equal to lớn one another. Sincex2+(10/3)x+(25/9) = 16/9 andx2+(10/3)x+(25/9) = (x+(5/3))2 then, according to the law of transitivity,(x+(5/3))2 = 16/9We"ll refer lớn this Equation as Eq. #6.2.1 The Square Root Principle says that When two things are equal, their square roots are equal.Note that the square root of(x+(5/3))2 is(x+(5/3))2/2=(x+(5/3))1=x+(5/3)Now, applying the Square Root Principle lớn Eq.#6.2.1 we get:x+(5/3)= √ 16/9 Subtract 5/3 from both sides lớn obtain:x = -5/3 + √ 16/9 Since a square root has two values, one positive & the other negativex2 + (10/3)x + 1 = 0has two solutions:x = -5/3 + √ 16/9 orx = -5/3 - √ 16/9 note that √ 16/9 can be written as√16 / √9which is 4 / 3

### Solve Quadratic Equation using the Quadratic Formula

6.3Solving3x2+10x+3 = 0 by the Quadratic Formula.According khổng lồ the Quadratic Formula,x, the solution forAx2+Bx+C= 0 , where A, B & C are numbers, often called coefficients, is given by :-B± √B2-4ACx = ————————2A In our case,A= 3B= 10C= 3 Accordingly,B2-4AC=100 - 36 =64Applying the quadratic formula : -10 ± √ 64 x=—————6Can √ 64 be simplified ?Yes!The prime factorization of 64is2•2•2•2•2•2 to be able to lớn remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. Second root).√ 64 =√2•2•2•2•2•2 =2•2•2•√ 1 =±8 •√ 1 =±8 So now we are looking at:x=(-10±8)/6Two real solutions:x =(-10+√64)/6=(-5+4)/3= -0.333 or:x =(-10-√64)/6=(-5-4)/3= -3.000