Math  /  Algebra

QuestionUse Synthetic Division to determine each quotient (and remainder - if necessary). If possible, use the quotients to write the dividend in factored form and list all the zeros. If NOT possible with given divisor, write DNE for all applicable answers. Complete work must be shown for credit! (x45x2+4)÷(x+2)\left(x^{4}-5 x^{2}+4\right) \div(x+2)
Quotient: \square Factored Form: \square Zeros: Select an answer \vee \square

Studdy Solution

STEP 1

What is this asking? We're diving a polynomial x45x2+4x^4 - 5x^2 + 4 by x+2x + 2 using synthetic division, then using the result to find the *factored form* of the original polynomial and its *zeros*. Watch out! Don't forget to use **zero** as the coefficient for any missing terms in the dividend!
Also, remember that synthetic division gives you the coefficients of the quotient, but you need to figure out the correct powers of xx yourself.

STEP 2

1. Set up the synthetic division.
2. Perform the synthetic division.
3. Write the quotient and remainder.
4. Factor the quotient.
5. Find the zeros.

STEP 3

Our **dividend** is x45x2+4x^4 - 5x^2 + 4 and our **divisor** is x+2x + 2.

STEP 4

Notice that the dividend is missing the x3x^3 and the xx terms!
We need to rewrite it with **zero coefficients** for these missing terms: x4+0x35x2+0x+4x^4 + 0x^3 - 5x^2 + 0x + 4.
Now we can grab the coefficients: **1**, **0**, **-5**, **0**, and **4**.

STEP 5

Since the divisor is x+2x + 2, we'll use 2-2 in our synthetic division (think: what makes x+2x+2 equal to zero?).

STEP 6

We write the 2-2 outside the box and the coefficients of the dividend inside: 210504\begin{array}{cccccc} -2 & 1 & 0 & -5 & 0 & 4 \\ \end{array}

STEP 7

Bring down the **leading coefficient**, which is **1**, below the line. 2105041\begin{array}{cccccc} -2 & 1 & 0 & -5 & 0 & 4 \\ & 1 \end{array}

STEP 8

Multiply 2-2 by **1** to get 2-2, and place it under the next coefficient, **0**.
Add **0** and 2-2 to get 2-2, and write the result below the line. 21050412\begin{array}{cccccc} -2 & 1 & 0 & -5 & 0 & 4 \\ 1 & -2 \end{array} Repeat this process: multiply 2-2 by 2-2 to get **4**, add to 5-5 to get 1-1.
Multiply 2-2 by 1-1 to get **2**, add to **0** to get **2**.
Multiply 2-2 by **2** to get 4-4, add to **4** to get **0**. 21050412120\begin{array}{cccccc} -2 & 1 & 0 & -5 & 0 & 4 \\ 1 & -2 & -1 & 2 & 0 \end{array}

STEP 9

The last number, **0**, is the **remainder**.
The other numbers, **1**, 2-2, 1-1, and **2**, are the coefficients of the **quotient**.
Since we started with x4x^4, the quotient starts with x3x^3.

STEP 10

So, the **quotient** is x32x2x+2x^3 - 2x^2 - x + 2 and the **remainder** is **0**.

STEP 11

We can factor the quotient by grouping: x2(x2)(x2)=(x21)(x2)=(x1)(x+1)(x2)x^2(x - 2) - (x - 2) = (x^2 - 1)(x - 2) = (x - 1)(x + 1)(x - 2)

STEP 12

Since the remainder is zero, x+2x+2 is a factor.
So, the factored form of x45x2+4x^4 - 5x^2 + 4 is (x+2)(x1)(x+1)(x2)(x + 2)(x - 1)(x + 1)(x - 2).

STEP 13

Set each factor equal to zero and solve for xx: x+2=0x=2x + 2 = 0 \Rightarrow x = -2 x1=0x=1x - 1 = 0 \Rightarrow x = 1x+1=0x=1x + 1 = 0 \Rightarrow x = -1x2=0x=2x - 2 = 0 \Rightarrow x = 2

STEP 14

The zeros are 2-2, 11, 1-1, and 22.

STEP 15

Quotient: x32x2x+2x^3 - 2x^2 - x + 2 Factored Form: (x+2)(x1)(x+1)(x2)(x + 2)(x - 1)(x + 1)(x - 2) Zeros: 2-2, 11, 1-1, 22

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