Question

4x^4-41x^2+ 100 can be factorized as

Answer 1

Answer:

There is your answer↓

Step-by-step explanation:

STEP

1

:

Equation at the end of step 1

((4 • (x4)) - 41x2) + 100 = 0

STEP

2

:

Equation at the end of step

2

:

(22x4 - 41x2) + 100 = 0

STEP

3

:

Trying to factor by splitting the middle term

3.1 Factoring 4x4-41x2+100

The first term is, 4x4 its coefficient is 4 .

The middle term is, -41x2 its coefficient is -41 .

The last term, "the constant", is +100

Step-1 : Multiply the coefficient of the first term by the constant 4 • 100 = 400

Step-2 : Find two factors of 400 whose sum equals the coefficient of the middle term, which is -41 .

-400 + -1 = -401

-200 + -2 = -202

-100 + -4 = -104

-80 + -5 = -85

-50 + -8 = -58

-40 + -10 = -50

-25 + -16 = -41 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -25 and -16

4x4 - 25x2 - 16x2 - 100

Step-4 : Add up the first 2 terms, pulling out like factors :

x2 • (4x2-25)

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

4 • (4x2-25)

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

(x2-4) • (4x2-25)

Which is the desired factorization

Trying to factor as a Difference of Squares:

3.2 Factoring: 4x2-25

Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)

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 : 4 is the square of 2

Check : 25 is the square of 5

Check : x2 is the square of x1

Factorization is : (2x + 5) • (2x - 5)

Trying to factor as a Difference of Squares:

3.3 Factoring: x2 - 4

Check : 4 is the square of 2

Check : x2 is the square of x1

Factorization is : (x + 2) • (x - 2)

Equation at the end of step

3

:

(2x + 5) • (2x - 5) • (x + 2) • (x - 2) = 0

STEP

4

:

Theory - Roots of a product

4.1 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 separately

In other words, we are going to solve as many equations as there are terms in the product

Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation:

4.2 Solve : 2x+5 = 0

Subtract 5 from both sides of the equation :

2x = -5

Divide both sides of the equation by 2:

x = -5/2 = -2.500

Solving a Single Variable Equation:

4.3 Solve : 2x-5 = 0

Add 5 to both sides of the equation :

2x = 5

Divide both sides of the equation by 2:

x = 5/2 = 2.500

Solving a Single Variable Equation:

4.4 Solve : x+2 = 0

Subtract 2 from both sides of the equation :

x = -2

Solving a Single Variable Equation:

4.5 Solve : x-2 = 0

Add 2 to both sides of the equation :

x = 2

Supplement : Solving Quadratic Equation Directly

Solving 4x4-41x2+100 = 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 Formula

Solving a Single Variable Equation:

Equations which are reducible to quadratic :

5.1 Solve 4x4-41x2+100 = 0

This equation is reducible to 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 :

4w2-41w+100 = 0

Solving this new equation using the quadratic formula we get two real solutions :

6.2500 or 4.0000

Now 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

4x4-41x2+100 = 0

are either :

x =√ 6.250 = 2.50000 or :

x =√ 6.250 = -2.50000 or :

x =√ 4.000 = 2.00000 or :

x =√ 4.000 = -2.00000

I think this is helpful for you

Answer 2

Step-by-step explanation:

$4 {x}^{4} - 41 {x}^{2} + 100 \\ = {(2 {x}^{2} )}^{2} - 2 \times 2 {x}^{2} \times 10 + {10}^{2} \\ = {(2x - 10)}^{2} \\ = (2x - 10)(2x - 10) \\ \\ \\ \\ hope \: it \: helps \: you......... \\ please \: mark \: my \: answer \: as \: \\ brainleist........$