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

A chef at a restaurant uses 17 pounds of butter each day. About how many grams does the chef use each day?

24% of 72 is what number? (will mark brainliest for full explanation)

helpppp it’s my friends birthday

Choose the correct expression for 10 decreased by 3x

Solve for w.

w−8=32


w = 4

w = 24

w = 40

w = 256

ANSWER QUICKLY!!!11!!!1!1

what is 3/4 divided by one - sixths

Use the chart and graph to find the number of pairs of size 10 1/2 shoes the store orders for a total order of 175 pairs of shoes.

Someone please help me with number 6 lve been stuck on it forever

a water park is designing an incline for guests to boogie board on as shown below
PLEASE HELP

9. Which expression is equivalent to 14(8x−36)+2x? (1 point)

a. 10x - 36

b. 30x

c. 4x - 9

d. 4x - 36

Stands steakhouse has a server to cook ratio of 5 to to tell a number of servers and cooks are 42 how many servers the stands steakhouse employ

28 is 7 times what number?
8 is 32 times what number?
4000 is 4 times what number?
Choose one part and explain how you know your answer is correct.

What is an equation in point-slope form of the line that passes through (1, −6) and (4, 6)?

On a test a student got 15 problems correct out of a total of 20 problems.
What percent did the student get incorrect?
HURRY

The point-slope form of the equation of a line that passes through points (8, 4) and (0, 2) is y - 4 = (x – 8). What
is the slope-intercept form of the equation for this line?
O y = 1 x - 12
O y = -x-4
O y=+*+2
O y=x+6

hhhhhhhhhhhhhhhhhelp hhhhhhhhhhhhhhhhhelp hhhhhhhhhhhhhhhhhelp hhhhhhhhhhhhhhhhhelp hhhhhhhhhhhhhhhhhelp

W-4y+z=0 for z (please show work)

what is difference between applied force and normal force?​

help ⁉️⁉️⁉️⁉️⁉️⁉️⁉️⁉️⁉️⁉️⁉️⁉️⁉️⁉️⁉️⁉️⁉️⁉️⁉️⁉️ 3) Scott is twice as old as Ben. Donna is 6 years older than Ben. The sum of their ages is 34. How old are Donna, Scott, and Ben?​

is 0.75 repeating irrational or rational

Of the 16 children at the park, 10 are riding bicycles and 6 are riding skateboards. Select the
statements for which the ratio is correct.
a. The ratio of bicycles to skateboards is 10 to 6
b. The ratio of skateboards to children is 3.8
c. The ratio of bicycles to children 6 to 16
d. The ratio of skateboards to children is 6 10
The ratio of bicycles to children is 5,8
The ratio of bicycles to skateboards is 5.3
e
f.
f

Please help !! Which are pairs of vertical angles?


< AOC and

A company pays 11% commission for $10,000.00 or less and 14% commission for over
$10,000.00. What is the commission for sales of $18,236.75?

Todd and Kyle wrote down these numbers. Todd’s number is 78,452 Kyle’s number is 94,725 Which statement below is true about the values of the numbers? *

The value of the 2 in Todd’s number is 10 times larger than the value of the 2 in Kyle's number.

The value of the 5 in Kyle’s number is 10 times larger than the value of the 5 in Todd's number.

The value of the 4 in Kyle’s number is 10 times larger than the value of the 4 in Todd’s number.

The value of the 4 in Todd's number is 10 times larger than the value of the 4 in Kyle’s number.

What is the polynomial of least degree with the given zeroes O with a
multiplicity of 3, -1 with a multiplicity of 2, i, and -i?

Choose the correct rule for the graph shown. Please Help 30POINTS

How many gallons is equivalent to 20 quarts. There are 4 quarts in a gallon.
Label your answer in gallons.

A line passes through the points (8, 1) and (6, 7). What is the slope?

Write 7 more than w is 25 as an algebraic equation.

In right ∆ABC with m∠B=30°, AC = 4.
Find AB, HB and Area of triangle ABC