# The Elements Of Written Arithmetic: Combining Analysis And Synthesis

### (x-1)(x+3)(x-2)(x-6)+36=0

This solution deals with finding the roots (zeroes) of polynomials.

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## Step by Step Solution

## Step by step solution :

## Step 1 :

Equation at the end of step 1 : (((x-1)•(x+3)•(x-2))•(x-6))+36 = 0## Step 2 :

Equation at the over of step 2 : ((x-1)•(x+3)•(x-2)•(x-6))+36 = 0## Step 3 :

Equation at the end of step 3 : (x-1)•(x+3)•(x-2)•(x-6)+36 = 0## Step 4 :

## Step 5 :

Pulling out lượt thích terms :5.1 Pull out lượt thích factors:x4 - 6x3 - 7x2 + 48x=x•(x3 - 6x2 - 7x + 48)Checking for a perfect cube :5.2x3 - 6x2 - 7x + 48 is not a perfect cube

Trying to factor by pulling out :5.3 Factoring: x3 - 6x2 - 7x + 48 Thoughtfully split the expression at hand into groups, each group having two terms:Group 1: -7x + 48Group 2: -6x2 + x3Pull out from each group separately :Group 1: (-7x + 48) • (1) = (7x - 48) • (-1)Group 2: (x - 6) • (x2)Bad news !! Factoring by pulling out fails : The groups have no common factor và can not be added up to khung a multiplication.

### Polynomial Roots Calculator :

5.4 Find roots (zeroes) of : F(x) = x3 - 6x2 - 7x + 48Polynomial Roots Calculator is a mix of methods aimed at finding values ofxfor 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 integersThe Rational Root Theorem states that if a polynomial zeroes for a rational numberP/Q then phường is a factor of the Trailing Constant và Q is a factor of the Leading CoefficientIn this case, the Leading Coefficient is 1 & the Trailing Constant is 48. The factor(s) are: of the Leading Coefficient : 1of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,8 ,12 ,16 ,24 ,48 Let us chạy thử ....

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-1 | 1 | -1.00 | 48.00 | ||||||

-2 | 1 | -2.00 | 30.00 | ||||||

-3 | 1 | -3.00 | -12.00 | ||||||

-4 | 1 | -4.00 | -84.00 | ||||||

-6 | 1 | -6.00 | -342.00 | ||||||

-8 | 1 | -8.00 | -792.00 | ||||||

-12 | 1 | -12.00 | -2460.00 | ||||||

-16 | 1 | -16.00 | -5472.00 | ||||||

-24 | 1 | -24.00 | -17064.00 | ||||||

-48 | 1 | -48.00 | -124032.00 | ||||||

1 | 1 | 1.00 | 36.00 | ||||||

2 | 1 | 2.00 | 18.00 | ||||||

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

4 | 1 | 4.00 | -12.00 | ||||||

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

8 | 1 | 8.00 | 120.00 | ||||||

12 | 1 | 12.00 | 828.00 | ||||||

16 | 1 | 16.00 | 2496.00 | ||||||

24 | 1 | 24.00 | 10248.00 | ||||||

48 | 1 | 48.00 | 96480.00 |

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 to lớn its lowest terms In our case this means that x3 - 6x2 - 7x + 48can be divided with x - 3

### Polynomial Long Division :

5.5 Polynomial Long Division Dividing : x3 - 6x2 - 7x + 48("Dividend") By:x - 3("Divisor")

dividend | x3 | - | 6x2 | - | 7x | + | 48 | ||

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

remainder | - | 3x2 | - | 7x | + | 48 | |||

-divisor | * -3x1 | - | 3x2 | + | 9x | ||||

remainder | - | 16x | + | 48 | |||||

-divisor | * -16x0 | - | 16x | + | 48 | ||||

remainder | 0 |

Quotient : x2-3x-16 Remainder: 0

Trying to factor by splitting the middle term5.6Factoring x2-3x-16 The first term is, x2 its coefficient is 1.The middle term is, -3x its coefficient is -3.The last term, "the constant", is -16Step-1 : Multiply the coefficient of the first term by the constant 1•-16=-16Step-2 : Find two factors of -16 whose sum equals the coefficient of the middle term, which is -3.

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-16 | + | 1 | = | -15 | ||

-8 | + | 2 | = | -6 | ||

-4 | + | 4 | = | 0 | ||

-2 | + | 8 | = | 6 | ||

-1 | + | 16 | = | 15 |

Observation : No two such factors can be found !! Conclusion : Trinomial can not be factored

Equation at the over of step 5 :x • (x2 - 3x - 16) • (x - 3) = 0

## Step 6 :

Theory - Roots of a sản phẩm :6.1 A hàng hóa 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 lớn solve as many equations as there are terms in the productAny solution of term = 0 solves hàng hóa = 0 as well.Solving a Single Variable Equation:6.2Solve:x = 0 Solution is x = 0

Parabola, Finding the Vertex:6.3Find the Vertex ofy = x2-3x-16Parabolas 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 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 khổng lồ be able to 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 1.5000Plugging into the parabola formula 1.5000 for x we can calculate the y-coordinate:y = 1.0 * 1.50 * 1.50 - 3.0 * 1.50 - 16.0 or y = -18.250

Parabola, Graphing Vertex và X-Intercepts :Root plot for : y = x2-3x-16 Axis of Symmetry (dashed) x= 1.50 Vertex at x,y = 1.50,-18.25 x-Intercepts (Roots) : Root 1 at x,y = -2.77, 0.00 Root 2 at x,y = 5.77, 0.00

Solve Quadratic Equation by Completing The Square6.4Solvingx2-3x-16 = 0 by Completing The Square.Add 16 to both side of the equation : x2-3x = 16Now the clever bit: Take the coefficient of x, which is 3, divide by two, giving 3/2, & finally square it giving 9/4Add 9/4 khổng lồ both sides of the equation :On the right hand side we have:16+9/4or, (16/1)+(9/4)The common denominator of the two fractions is 4Adding (64/4)+(9/4) gives 73/4So adding to both sides we finally get:x2-3x+(9/4) = 73/4Adding 9/4 has completed the left hand side into a perfect square :x2-3x+(9/4)=(x-(3/2))•(x-(3/2))=(x-(3/2))2 Things which are equal khổng lồ the same thing are also equal lớn one another. Sincex2-3x+(9/4) = 73/4 andx2-3x+(9/4) = (x-(3/2))2 then, according khổng lồ the law of transitivity,(x-(3/2))2 = 73/4We"ll refer to lớn this Equation as Eq. #6.4.1 The Square Root Principle says that When two things are equal, their square roots are equal.Note that the square root of(x-(3/2))2 is(x-(3/2))2/2=(x-(3/2))1=x-(3/2)Now, applying the Square Root Principle to lớn Eq.#6.4.1 we get:x-(3/2)= √ 73/4 địa chỉ cửa hàng 3/2 to lớn both sides to obtain:x = 3/2 + √ 73/4 Since a square root has two values, one positive và the other negativex2 - 3x - 16 = 0has two solutions:x = 3/2 + √ 73/4 orx = 3/2 - √ 73/4 note that √ 73/4 can be written as√73 / √4which is √73 / 2

### Solve Quadratic Equation using the Quadratic Formula

6.5Solvingx2-3x-16 = 0 by the Quadratic Formula.According to lớn the Quadratic Formula,x, the solution forAx2+Bx+C= 0 , where A, B và C are numbers, often called coefficients, is given by :-B± √B2-4ACx = ————————2A In our case,A= 1B= -3C=-16 Accordingly,B2-4AC=9 - (-64) = 73Applying the quadratic formula : 3 ± √ 73 x=—————2 √ 73 , rounded khổng lồ 4 decimal digits, is 8.5440So now we are looking at:x=(3± 8.544 )/2Two real solutions:x =(3+√73)/2= 5.772 or:x =(3-√73)/2=-2.772

Solving a Single Variable Equation:6.6Solve:x-3 = 0Add 3 to lớn both sides of the equation:x = 3