Math / Algebra

QuestionUse synthetic division to determine if the given value for $k$ is a zero of this polynomial. If not, determine $p(k)$ . $p(x)=4 x^{4}-19 x^{3}-9 x^{2}+98 x-24 ; k=3$

Studdy Solution

STEP 1

What is this asking? Is $3$ a root of the polynomial $4x^4 - 19x^3 - 9x^2 + 98x - 24$ ?
If not, what's the polynomial's value at $x = 3$ ? Watch out! Remember, a value $k$ is a zero of a polynomial if the remainder after synthetic division by $x-k$ is zero!
Don't mix up the remainder with the result.

STEP 2

1. Set up the synthetic division
2. Perform the synthetic division
3. Interpret the result

STEP 3

Alright, let's set up our synthetic division!
We'll put our potential root, $k = 3$ , outside the box.
Inside, we'll list the coefficients of our polynomial $p(x) = 4x^4 - 19x^3 - 9x^2 + 98x - 24$ , which are $4$ , $-19$ , $-9$ , $98$ , and $-24$ .

STEP 4

So, it looks like this:
$\begin{array}{cccccc} 3 & 4 & -19 & -9 & 98 & -24 \\ \end{array}$

STEP 5

Bring down the leading coefficient, $4$ , below the line.
$\begin{array}{cccccc} 3 & 4 & -19 & -9 & 98 & -24 \\ & 4 & & & & \end{array}$

STEP 6

Multiply $3$ by $4$ to get $3 \cdot 4 = 12$ , and place it under the next coefficient, $-19$ .
$\begin{array}{ccccc} 3 & 4 & -19 & -9 & 98 & -24 \\ 4 & 12 \end{array}$

STEP 7

Add $-19$ and $12$ to get $-19 + 12 = -7$ , and write the result below the line.
$\begin{array}{cccccc} 3 & 4 & -19 & -9 & 98 & -24 \\ 4 & -7 \end{array}$

STEP 8

Repeat this process: multiply $3$ by $-7$ to get $-21$ , add it to $-9$ to get $-30$ , multiply by $3$ to get $-90$ , add it to $98$ to get $8$ , multiply by $3$ to get $24$ , and finally, add it to $-24$ to get $0$ .
$\begin{array}{cccccc} 3 & 4 & -19 & -9 & 98 & -24 \\ 4 & -7 & -30 & 8 & 0 \end{array}$

STEP 9

That last number, $0$ , is our remainder!
Since the remainder is $0$ , we know that $k = 3$ is a zero of the polynomial!
Woohoo!

STEP 10

$3$ is a zero of the polynomial $p(x) = 4x^4 - 19x^3 - 9x^2 + 98x - 24$ , as the remainder after synthetic division is $0$ .

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QuestionUse synthetic division to determine if the given value for $k$ is a zero of this polynomial. If not, determine $p(k)$ . $p(x)=4 x^{4}-19 x^{3}-9 x^{2}+98 x-24 ; k=3$

Studdy Solution

STEP 1

STEP 2

1. Set up the synthetic division
2. Perform the synthetic division
3. Interpret the result

STEP 3

Alright, let's set up our synthetic division!
We'll put our potential root, $k = 3$ , outside the box.
Inside, we'll list the coefficients of our polynomial $p(x) = 4x^4 - 19x^3 - 9x^2 + 98x - 24$ , which are $4$ , $-19$ , $-9$ , $98$ , and $-24$ .

STEP 4

So, it looks like this:
$\begin{array}{cccccc} 3 & 4 & -19 & -9 & 98 & -24 \\ \end{array}$

STEP 5

Bring down the leading coefficient, $4$ , below the line.
$\begin{array}{cccccc} 3 & 4 & -19 & -9 & 98 & -24 \\ & 4 & & & & \end{array}$

STEP 6

Multiply $3$ by $4$ to get $3 \cdot 4 = 12$ , and place it under the next coefficient, $-19$ .
$\begin{array}{ccccc} 3 & 4 & -19 & -9 & 98 & -24 \\ 4 & 12 \end{array}$

STEP 7

Add $-19$ and $12$ to get $-19 + 12 = -7$ , and write the result below the line.
$\begin{array}{cccccc} 3 & 4 & -19 & -9 & 98 & -24 \\ 4 & -7 \end{array}$

STEP 8

STEP 9

That last number, $0$ , is our remainder!
Since the remainder is $0$ , we know that $k = 3$ is a zero of the polynomial!
Woohoo!

STEP 10

$3$ is a zero of the polynomial $p(x) = 4x^4 - 19x^3 - 9x^2 + 98x - 24$ , as the remainder after synthetic division is $0$ .

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QuestionUse synthetic division to determine if the given value for kkk is a zero of this polynomial. If not, determine p(k)p(k)p(k). p(x)=4x4−19x3−9x2+98x−24;k=3p(x)=4 x^{4}-19 x^{3}-9 x^{2}+98 x-24 ; k=3p(x)=4x4−19x3−9x2+98x−24;k=3

Studdy Solution

STEP 1

STEP 2

1. Set up the synthetic division2. Perform the synthetic division3. Interpret the result

STEP 3

STEP 4

So, it looks like this:34−19−998−24\begin{array}{cccccc} 3 & 4 & -19 & -9 & 98 & -24 \\ \end{array}3​4​−19​−9​98​−24​

STEP 5

Bring down the **leading coefficient**, 444, below the line.34−19−998−244\begin{array}{cccccc} 3 & 4 & -19 & -9 & 98 & -24 \\ & 4 & & & & \end{array}3​44​−19​−9​98​−24​

STEP 6

**Multiply** 333 by 444 to get 3⋅4=123 \cdot 4 = 123⋅4=12, and place it under the next coefficient, −19-19−19.34−19−998−24412\begin{array}{ccccc} 3 & 4 & -19 & -9 & 98 & -24 \\ 4 & 12 \end{array}34​412​−19−998−24

STEP 7

**Add** −19-19−19 and 121212 to get −19+12=−7-19 + 12 = -7−19+12=−7, and write the result below the line.34−19−998−244−7\begin{array}{cccccc} 3 & 4 & -19 & -9 & 98 & -24 \\ 4 & -7 \end{array}34​4−7​−19−998−24

STEP 8

STEP 9

That **last number**, 000, is our **remainder**!Since the remainder is 000, we know that k=3k = 3k=3 *is* a zero of the polynomial!Woohoo!

STEP 10

333 is a zero of the polynomial p(x)=4x4−19x3−9x2+98x−24p(x) = 4x^4 - 19x^3 - 9x^2 + 98x - 24p(x)=4x4−19x3−9x2+98x−24, as the remainder after synthetic division is 000.

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QuestionUse synthetic division to determine if the given value for $k$ is a zero of this polynomial. If not, determine $p(k)$ . $p(x)=4 x^{4}-19 x^{3}-9 x^{2}+98 x-24 ; k=3$

1. Set up the synthetic division
2. Perform the synthetic division
3. Interpret the result

So, it looks like this:
$\begin{array}{cccccc} 3 & 4 & -19 & -9 & 98 & -24 \\ \end{array}$

Bring down the leading coefficient, $4$ , below the line.
$\begin{array}{cccccc} 3 & 4 & -19 & -9 & 98 & -24 \\ & 4 & & & & \end{array}$

Multiply $3$ by $4$ to get $3 \cdot 4 = 12$ , and place it under the next coefficient, $-19$ .
$\begin{array}{ccccc} 3 & 4 & -19 & -9 & 98 & -24 \\ 4 & 12 \end{array}$

Add $-19$ and $12$ to get $-19 + 12 = -7$ , and write the result below the line.
$\begin{array}{cccccc} 3 & 4 & -19 & -9 & 98 & -24 \\ 4 & -7 \end{array}$

That last number, $0$ , is our remainder!
Since the remainder is $0$ , we know that $k = 3$ is a zero of the polynomial!
Woohoo!

$3$ is a zero of the polynomial $p(x) = 4x^4 - 19x^3 - 9x^2 + 98x - 24$ , as the remainder after synthetic division is $0$ .