Obtain all the other zeros of x⁴ 7x³ 5x² +35x 50 if two of its zeros are √ 5 and √ 5

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Step by step solution :
Step 1 :
Step 3 :
Step 4 :
Step 5 :
Three solutions were found :

This solution deals with simplification or other simple results.

Simplification or other simple results

((0 - 5x2) - 35x) - 50

3.1 Pull out like factors :

-5x2 - 35x - 50 = -5 • (x2 + 7x + 10)

Trying to factor by splitting the middle term

3.2 Factoring x2 + 7x + 10

The first term is, x2 its coefficient is 1 .

The middle term is, +7x its coefficient is 7 .
The last term, "the constant", is +10

Step-1 : Multiply the coefficient of the first term by the constant 1 • 10 = 10

Step-2 : Find two factors of 10 whose sum equals the coefficient of the middle term, which is 7 .

-10	+	-1	=	-11
-5	+	-2	=	-7
-2	+	-5	=	-7
-1	+	-10	=	-11
1	+	10	=	11
2	+	5	=	7	That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 2 and 5
x2 + 2x + 5x + 10Step-4 : Add up the first 2 terms, pulling out like factors :

x • (x+2)

Add up the last 2 terms, pulling out common factors :

5 • (x+2)

Step-5 : Add up the four terms of step 4 :

(x+5) • (x+2)

Which is the desired factorization

Final result :

-5 • (x + 5) • (x + 2)

Changes made to your input should not affect the solution:

(1): "x5" was replaced by "x^5".

Step by step solution :

Step 1 :

Equation at the end of step 1 :

(5 • (x2)) - (5•7x50) = 0

Equation at the end of step 2 :

5x2 - (5•7x50) = 0

Step 3 :

Step 4 :

Pulling out like terms :

4.1 Pull out like factors :

5x2 - 35x50 = -5x2 • (7x48 - 1)

Trying to factor as a Difference of Squares :

4.2 Factoring: 7x48 - 1

Theory : A difference of two perfect squares, A2 - B2 can be factored into Proof :

  (A+B) • (A-B) =
         A2 - AB + BA - B2 =
         A2 - AB + AB - B2 =
         A2 - B2

Note : AB = BA is the commutative property of multiplication.

Note : - AB + AB equals zero and is therefore eliminated from the expression.

Check :Ruling : Binomial can not be factored as the

difference of two perfect squares

Trying to factor as a Difference of Cubes:

4.3 Factoring: 7x48 - 1

Theory : A difference of two perfect cubes, a3 - b3 can be factored into

(a-b) • (a2 +ab +b2)

Proof :  (a-b)•(a2+ab+b2) =
            a3+a2b+ab2-ba2-b2a-b3 =
            a3+(a2b-ba2)+(ab2-b2a)-b3 =
            a3+0+0-b3 =
            a3-b3

Check : 7 is not a cube !!

Ruling : Binomial can not be factored as the difference of two perfect cubes

Equation at the end of step 4 :

-5x2 • (7x48 - 1) = 0

Step 5 :

Theory - Roots of a product :

5.1 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 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 :

5.2 Solve : -5x2 = 0Multiply both sides of the equation by (-1) : 5x2 = 0 Divide both sides of the equation by 5:

x2 = 0

When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:
x = ± √ 0 Any root of zero is zero. This equation has one solution which is x = 0