Question

Math

Posted 6 months ago

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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:
```

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Answer from Sia

Posted 6 months ago

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.

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