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)
