Question

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

(0, 4)
(–1, 0) and (1, 0)
(–1, 0), (0, 4), and (1, 0)
(–4, 0), (–1, 0), (0, 4), and (1, 0)

in case you are wondering, there is no graph.

Answer 1

`4(x + 1)^(2)(x - 1)^(2)`

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