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

Quotient
Factored Form:
Zeros:

Question

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

Quotient
Factored Form:
Zeros:

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To determine the quotient using Synthetic Division, we start with the polynomial $x^4 - 5x^2 + 4$ and divide it by $x + 2$. ‖ ‖ step 2 ⋮ Set up the synthetic division by writing the coefficients of $x^4 - 5x^2 + 4$: $[1, 0, -5, 0, 4]$. ‖ ‖ step 3 ⋮ Write the divisor $x + 2$ as $-2$ and perform the synthetic division: ‖ ‖ step 4 ⋮ $$ \begin{array}{r|rrrrr} -2 & 1 & 0 & -5 & 0 & 4 \ & & -2 & 4 & 2 & -4 \ \hline & 1 & -2 & -1 & 2 & 0 \ \end{array} $$ The quotient is $x^3 - 2x^2 - x + 2$ and the remainder is $0$. ‖ ‖ step 5 ⋮ Since the remainder is $0$, we can write the polynomial as $x^4 - 5x^2 + 4 = (x^3 - 2x^2 - x + 2)(x + 2)$. ‖ ‖ step 6 ⋮ To factor $x^4 - 5x^2 + 4$ completely, we factor $x^3 - 2x^2 - x + 2$ further: ‖ ‖ step 7 ⋮ $$ x^3 - 2x^2 - x + 2 = (x - 1)(x + 1)(x - 2) $$ Thus, $$ x^4 - 5x^2 + 4 = (x - 1)(x + 1)(x - 2)(x + 2) $$ ‖ ‖ step 8 ⋮ The zeros of the polynomial are the solutions to $(x - 1)(x + 1)(x - 2)(x + 2) = 0$, which are $x = 1, -1, 2, -2$. ‖ ∻Answer∻ ⚹ Quotient: $x^3 - 2x^2 - x + 2$ ⚹ ⚹ Factored Form: $(x - 1)(x + 1)(x - 2)(x + 2)$ ⚹ ⚹ Zeros: $x = 1, -1, 2, -2$ ⚹ ∻Key Concept∻ ⚹Synthetic Division⚹ ∻Explanation∻ ⚹Synthetic Division is a simplified method of dividing a polynomial by a binomial of the form $x - c$. It is particularly useful for finding the quotient and remainder quickly.⚹◊What is the process for using Synthetic Division to determine each quotient of the polynomial equation $(x^4 - 5x^2 + 4) ÷ (x + 2)$?⍭ Generate me a similar question◊

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