Mathematics, 03.07.2019 18:00, JamesLachoneus

Write the polynomial f(x)=x^4-10x^3+25x^2-40x+84. in factored form

Answers: 1

Show answers

Answers

answered: 1129682

Steps:

So firstly, to factor this we need to first find the potential roots of this polynomial. To find it, the equation is $\pm \frac{p}{q}$ , with p = the factors of the constant and q = the factors of the leading coefficient. In this case:

$\textsf{leading coefficient = 1, constant = 84}\\\\p=1,2,3,4,6,7,12,14,21,28,42,84\\q=1\\\\\pm \frac{1,2,3,4,6,7,12,14,21,28,42,84}{1}\\\\\textsf{Potential roots =}\pm 1, \pm 2,\pm 3,\pm 4,\pm 6, \pm 7,\pm 12,\pm 14,\pm 21,\pm 28,\pm 42,\pm 84$

Next, plug in the potential roots into x of the equation until one of them ends with a result of 0:

$f(1)=(1)^4-10(1)^3+25(1)^2-40(1)+84\\f(1)=1-10+25-40+84\\f(1)=60\ \textsf{Not a root}\\\\f(2)=2^4-10(2)^3+25(2)^2-40(2)+84\\f(2)=16-10*8+25*4-80+84\\f(2)=16-80+100-80+84\\f(2)=80\ \textsf{Not a root}\\\\f(3)=3^4-10(3)^3+25(3)^2-40(3)+84\\f(3)=81-10*27+25*9-120+84\\f(3)=81-270+225-120+84\\f(3)=0\ \textsf{Is a root}$

Since we know that 3 is a root, this means that one of the factors is (x - 3). Now that we know one of the roots, we are going to use synthetic division to divide the polynomial. To set it up, place the root of the divisor, in this case 3 from x - 3, on the left side and the coefficients of the original polynomial on the right side as such:

3 | 1 - 10 + 25 - 40 + 84_________________

Firstly, drop the 1:

3 | 1 - 10 + 25 - 40 + 84 ↓_________________ 1

Next, multiply 3 and 1, then add the product with -10:

3 | 1 - 10 + 25 - 40 + 84 ↓ + 3_________________ 1 - 7

Next, multiply 3 and -7, then add the product with 25:

3 | 1 - 10 + 25 - 40 + 84 ↓ + 3 - 21_________________ 1 - 7 + 4

Next, multiply 3 and 4, then add the product with -40:

3 | 1 - 10 + 25 - 40 + 84 ↓ + 3 - 21 + 12_________________ 1 - 7 + 4 - 28

Lastly, multiply -28 and 3, then add the product with 84:

3 | 1 - 10 + 25 - 40 + 84 ↓ + 3 - 21 + 12 - 84_________________ 1 - 7 + 4 - 28 + 0

Now our synthetic division is complete. Now since the degree of the original polynomial is 4, this means our quotient has a degree of 3 and follows the format ax^3+bx^2+cx+d . In this case, our quotient is x^3-7x^2+4x-28 .

So right now, our equation looks like this:

f(x)=(x-3)(x^3-7x^2+4x-28)

However, our second factor can be further simplified. For the second factor, I will be factoring by grouping. So factor x³ - 7x² and 4x - 28 separately. Make sure that they have the same quantity inside the parentheses:

f(x)=(x-3)(x^2(x-7)+4(x-7))

Now it can be rewritten as:

f(x)=(x-3)(x^2+4)(x-7)

Since the polynomial cannot be further simplified, your answer is:

f(x)=(x-3)(x^2+4)(x-7)

answered: Guest

answer:

your answer is: c. 196 7/8 in^3.

answered: Guest

its kinda hard to answer this when we dont have all the information that we need.

Other questions on the subject: Mathematics

Mathematics, 21.06.2019 15:00, lm18618

Jeffery conducted a survey in his school and found that 30 of the 50 eighth grade students' favorite subject is mathematics. based on the results, jeffery concluded that out of the 200 students in his school, 120 students' favorite subject is mathematics. select the statement that is true about jeffery's conclusion. a. jeffery's conclusion is not valid because the sample was biased since only 200 students were surveyed. b. jeffery's conclusion is valid because the sample was random since all of the students were eighth grade students. c. jeffery's conclusion is valid because the sample was random. d. jeffery's conclusion is not valid because the sample was biased since all of the students were eighth grade students.

Answers: 2

continue

Mathematics, 21.06.2019 17:40, nicole5918

Which of the following are accepted without proof in a logical system? check all that apply.

Answers: 1

continue

Mathematics, 21.06.2019 19:00, Jessieeeeey

Which shows the correct substitution of the values a, b and c from the equation 1=-2x+3x^2+1

Answers: 1

continue

Mathematics, 21.06.2019 19:30, zetrenne73

Jada has a meal in a restaurant she adds up the prices listed on the menu for everything they ordered and gets a subtotal of $42.00. when the check comes, it says they also need to pay $3.99 in sales tax. what percentage of the subtotal is the sales tax

Answers: 2

continue

Do you know the correct answer?

Write the polynomial f(x)=x^4-10x^3+25x^2-40x+84. in factored form...