Show all work to factor x^4 − 17x^2 + 16 completely.

I think I have to use completing the square or something but i don't know how ​

Answer:

[tex]x^{4}-17x^{2} +16[/tex] = (x -1)(x + 1)(x - 4)(x + 4)

Step-by-step explanation:

At first, let us find the first two factors of [tex]x^{4}-17x^{2} +16[/tex]

∵ The sign of the last term is positive

∴ The middle signs of the two factors are the same

∵ The sign of the middle term is negative

∴ The middle signs of the two factors are negative

∵ [tex]x^{4}[/tex] = x² × x² ⇒ first terms of the two factors

∵ 16 = -1 × -16 ⇒ second terms of the two factors

∵ x²(-1) + x²(-16) = -x² + -16x² = -17x² ⇒ the value of the middle term

∴ (x² - 1) and (x² - 16) are the factors of [tex]x^{4}-17x^{2} +16[/tex]

Now let us factorize each factor

→ The factors of the binomial a² - b² (difference of two squares) are

   (a - b) and (a + b)

∵ x² - 1 is the difference of two squares

∴ Its factors are (x - 1) and (x + 1)

∵ x² - 16 is the difference of two squares

∴ Its factors are (x - 4) and (x + 4)

∵ (x -1), (x + 1), (x - 4), and (x + 4) are the factors of (x² - 1) and (x² - 16)

∵ (x² - 1) and (x² - 16) are the factors of [tex]x^{4}-17x^{2} +16[/tex]

∴ (x -1), (x + 1), (x - 4), and (x + 4) are the factors of [tex]x^{4}-17x^{2} +16[/tex]

∴  [tex]x^{4}-17x^{2} +16[/tex] = (x -1)(x + 1)(x - 4)(x + 4)

Factoring an expression involves rewriting the expression in simpler forms

The factorized expression is (x - 4)(x + 4)(x - 1)(x + 1)

The expression is given as:

[tex]\mathbf{x^4 - 17x^2 + 16}[/tex]

Express 17 as 1 + 16

[tex]\mathbf{x^4 - 17x^2 + 16 = x^4 - x^2 - 16x^2 + 16}[/tex]

Factorize

[tex]\mathbf{x^4 - 17x^2 + 16 = x^2(x^2 - 1) - 16(x^2 - 1)}[/tex]

Factor out x^2 - 1

[tex]\mathbf{x^4 - 17x^2 + 16 =(x^2 - 16)(x^2 - 1)}[/tex]

Express 16 as 4^2, and 1 as 1^2

[tex]\mathbf{x^4 - 17x^2 + 16 =(x^2 - 4^2)(x^2 - 1^2)}[/tex]

Apply difference of two squares

[tex]\mathbf{x^4 - 17x^2 + 16 =(x - 4)(x + 4)(x - 1)(x + 1)}[/tex]

Hence, the factorized expression is (x - 4)(x + 4)(x - 1)(x + 1)

Read more about factoring expressions at:

https://brainly.com/question/19386208

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