Asksia AI LOGO

Sia

Question
Math
Posted 2 months ago

Use Synthetic Division to determine each quotient. Then use the quotients to write the dividend in factored form and list all the zeros. (x45x2+4)÷(x+2)\left(x^{4}-5 x^{2}+4\right) \div(x+2)

Quotient
Factored Form:
Zeros:
Sign in to unlock the answer
Answer from Sia
Posted 2 months ago
Solution by Steps
step 1
To determine the quotient using Synthetic Division, we start with the polynomial x45x2+4x^4 - 5x^2 + 4 and divide it by x+2x + 2
step 2
Set up the synthetic division by writing the coefficients of x45x2+4x^4 - 5x^2 + 4: [1,0,5,0,4][1, 0, -5, 0, 4]
step 3
Write the divisor x+2x + 2 as 2-2 and perform the synthetic division:
step 4
210504242412120 \begin{array}{r|rrrrr} -2 & 1 & 0 & -5 & 0 & 4 \\ & & -2 & 4 & 2 & -4 \\ \hline & 1 & -2 & -1 & 2 & 0 \\ \end{array} The quotient is x32x2x+2x^3 - 2x^2 - x + 2 and the remainder is 00
step 5
Since the remainder is 00, we can write the polynomial as x45x2+4=(x32x2x+2)(x+2)x^4 - 5x^2 + 4 = (x^3 - 2x^2 - x + 2)(x + 2)
step 6
To factor x45x2+4x^4 - 5x^2 + 4 completely, we factor x32x2x+2x^3 - 2x^2 - x + 2 further:
step 7
x32x2x+2=(x1)(x+1)(x2) x^3 - 2x^2 - x + 2 = (x - 1)(x + 1)(x - 2) Thus, x45x2+4=(x1)(x+1)(x2)(x+2) x^4 - 5x^2 + 4 = (x - 1)(x + 1)(x - 2)(x + 2)
step 8
The zeros of the polynomial are the solutions to (x1)(x+1)(x2)(x+2)=0(x - 1)(x + 1)(x - 2)(x + 2) = 0, which are x=1,1,2,2x = 1, -1, 2, -2
Answer
Quotient: x32x2x+2x^3 - 2x^2 - x + 2
Factored Form: (x1)(x+1)(x2)(x+2)(x - 1)(x + 1)(x - 2)(x + 2)
Zeros: x=1,1,2,2x = 1, -1, 2, -2
Key Concept
Synthetic Division
Explanation
Synthetic Division is a simplified method of dividing a polynomial by a binomial of the form xcx - c. It is particularly useful for finding the quotient and remainder quickly.

Not the question you are looking for? Ask here!

Enter question by text

Enter question by image

Upgrade to Asksia Pro

Join a AskSia's Pro Plan, and get 24/7 AI tutoring for your reviews, assignments, quizzes and exam preps.

Unlimited chat query usages
Strong algorithms that better know you
Early access to new release features
Study Other Question