Mathematics, 06.11.2020 03:10 blayneaafedt

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

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Answer from: chaseashley24

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:

$\mathbf{x^4 - 17x^2 + 16}$

Express 17 as 1 + 16

$\mathbf{x^4 - 17x^2 + 16 = x^4 - x^2 - 16x^2 + 16}$

Factorize

$\mathbf{x^4 - 17x^2 + 16 = x^2(x^2 - 1) - 16(x^2 - 1)}$

Factor out x^2 - 1

$\mathbf{x^4 - 17x^2 + 16 =(x^2 - 16)(x^2 - 1)}$

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

$\mathbf{x^4 - 17x^2 + 16 =(x^2 - 4^2)(x^2 - 1^2)}$

Apply difference of two squares

$\mathbf{x^4 - 17x^2 + 16 =(x - 4)(x + 4)(x - 1)(x + 1)}$

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

Read more about factoring expressions at:

link

Answer from: BeautyxQueen

$x^{4}-17x^{2} +16$ = (x -1)(x + 1)(x - 4)(x + 4)

Step-by-step explanation:

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

∵ 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

∵ $x^{4}$ = 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 $x^{4}-17x^{2} +16$

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 $x^{4}-17x^{2} +16$

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

∴ $x^{4}-17x^{2} +16$ = (x -1)(x + 1)(x - 4)(x + 4)

Answer from: Quest

There is no question, there is just the answer.

Answer from: Quest

$a_1 = 1$ , and

$a_{n+1} = a_n + (n + 1)$ for $n \ge 1$

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote $a_1 = 1$ , $a_2 = 3$ , $a_3 = 6$ , so that we can call

1 the 1st term, 3 the 2nd term, 6 the 3rd term,etc.

note the following:

from the 1st term (1) to the 2nd term (3), we go up by 2,from the 2nd term (3) to the 3rd term (6), we go up by 3from the 3rd term (6) to the 4th term (10), we go up by 4, andfrom the 4th term (10) to the 5th term (15), we go up by 5.

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from $a_1 = 1$ , to get to $a_2 = 3$ , we add $2(=1+1)$

from $a_2 = 3$ , to get to $a_3 = 6$ , we add $3 (=2+1)$

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

$a_1 = 1$ , and $a_{n+1} = a_n +(n + 1)$ for $n \ge 1$

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Show all work to factor x^4 − 17x^2 + 16 completely. I think I have to use completing the square or...

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