Question

Graph the function y = 4x4 – 8x2 + 4. Which lists all of the turning points of the graph?

Answer 1

Answer:

$4(x + 1)^{2}(x - 1)^{2}$

Step-by-step explanation:

STEP 1:

The equation at the end of step 1

$((4 (x^4)) - 2^3x^2) + 4$

STEP 2:

The equation at the end of step 2:

$(2^2x^4 - 2^3x^2) + 4$

STEP 3:

STEP 4: Pulling out like terms

4.1 Pull out like factors:

$4x^4 - 8x^2 + 4 = 4(x^4 - 2x^2 + 1)$

Trying to factor by splitting the middle term

4.2 Factoring $x^4 - 2x^2 + 1$

The first term is, $x^4$ its coefficient is 1.

The middle term is,  $-2x^2$  its coefficient is -2.

The last term, "the constant", is +1.

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

Step-2: Find two factors of  1  whose sum equals the coefficient of the middle term, which is -2.

-1 + -1 = -2 That's it

Step-3: Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -1  and  -1

                    x4 - 1x2 - 1x2 - 1

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

                   x2 • (x2-1)

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

                    1 • (x2-1)

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

                   (x2-1)  •  (x2-1)

            Which is the desired factorization

Trying to factor as a Difference of Squares:

4.3      Factoring:  x2-1

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 : 1 is the square of 1

Check :  x2  is the square of  x1

Factorization is :       (x + 1)  •  (x - 1)

Trying to factor as a Difference of Squares:

4.4      Factoring:  x2 - 1

Check : 1 is the square of 1

Check :  x2  is the square of  x1

Factorization is :       (x + 1)  •  (x - 1)

Multiplying Exponential Expressions:

4.5    Multiply  (x + 1)  by  (x + 1)

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (x+1)  and the exponents are :

         1 , as  (x+1)  is the same number as  (x+1)1

and   1 , as  (x+1)  is the same number as  (x+1)1

The product is therefore,  (x+1)(1+1) = (x+1)2

Multiplying Exponential Expressions:

4.6    Multiply  (x-1)  by  (x-1)

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (x-1)  and the exponents are :

         1 , as  (x-1)  is the same number as  (x-1)1

and   1 , as  (x-1)  is the same number as  (x-1)1

The product is therefore,  (x-1)(1+1) = (x-1)2

Final result :

 4 • (x + 1)2 • (x - 1)2

Answer 2

Answer:

STEP 1:

The equation at the end of step 1

STEP 2:

The equation at the end of step 2:

STEP 3:

STEP 4: Pulling out like terms

4.1 Pull out like factors:

Trying to factor by splitting the middle term

4.2 Factoring  

The first term is,  its coefficient is 1.

The middle term is,    its coefficient is -2.

The last term, "the constant", is +1.

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

Step-2: Find two factors of  1  whose sum equals the coefficient of the middle term, which is -2.

-1 + -1 = -2 That's it

Step-3: Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -1  and  -1

                   x4 - 1x2 - 1x2 - 1

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

                  x2 • (x2-1)

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

                   1 • (x2-1)

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

                  (x2-1)  •  (x2-1)

           Which is the desired factorization

Trying to factor as a Difference of Squares:

4.3      Factoring:  x2-1

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 : 1 is the square of 1

Check :  x2  is the square of  x1

Factorization is :       (x + 1)  •  (x - 1)

Trying to factor as a Difference of Squares:

4.4      Factoring:  x2 - 1

Check : 1 is the square of 1

Check :  x2  is the square of  x1

Factorization is :       (x + 1)  •  (x - 1)

Multiplying Exponential Expressions:

4.5    Multiply  (x + 1)  by  (x + 1)

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (x+1)  and the exponents are :

        1 , as  (x+1)  is the same number as  (x+1)1

and   1 , as  (x+1)  is the same number as  (x+1)1

The product is therefore,  (x+1)(1+1) = (x+1)2

Multiplying Exponential Expressions:

4.6    Multiply  (x-1)  by  (x-1)

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (x-1)  and the exponents are :

        1 , as  (x-1)  is the same number as  (x-1)1

and   1 , as  (x-1)  is the same number as  (x-1)1

The product is therefore,  (x-1)(1+1) = (x-1)2

Final result :

4 • (x + 1)2 • (x - 1)2

Step-by-step explanation: