Math / Algebra

Questiona. Use synthetic division to show that 2 is a solution of the polynomial equation below. $13 x^{3}+15 x^{2}-10 x-144=0$ b. Use the solution from part (a) to solve this problem. The number of eggs, $f(x)$ , in a female moth is a function of her abdominal width, in millimeters, modeled by the equation below. $f(x)=13 x^{3}+15 x^{2}-10 x-41$
What is the abdominal width when there are 103 eggs? a. The number 2 is a solution to the equation because the remainder of the division, $13 x^{3}+15 x^{2}-10 x-144$ divided by $x-2$ , is $\square$

Studdy Solution

STEP 1

1. We are given the polynomial $13x^3 + 15x^2 - 10x - 144 = 0$ .
2. We need to use synthetic division to verify that $x = 2$ is a solution.
3. We are given a function $f(x) = 13x^3 + 15x^2 - 10x - 41$ .
4. We need to find the abdominal width when there are 103 eggs, i.e., when $f(x) = 103$ .

STEP 2

1. Perform synthetic division of the polynomial $13x^3 + 15x^2 - 10x - 144$ by $x - 2$ .
2. Verify that the remainder is zero to confirm that $x = 2$ is a solution.
3. Use the result from part (a) to solve for $x$ in the equation $f(x) = 103$ .

STEP 3

Set up synthetic division for the polynomial $13x^3 + 15x^2 - 10x - 144$ with divisor $x - 2$ :
Write the coefficients: $13, 15, -10, -144$ .

STEP 4

Perform synthetic division:
1. Bring down the leading coefficient $13$ .
2. Multiply $13$ by $2$ and write the result under the next coefficient.
3. Add the result to the next coefficient $15$ .
4. Repeat the process for each coefficient.
$\begin{array}{r|rrrr} 2 & 13 & 15 & -10 & -144 \\ & & 26 & 82 & 144 \\ \hline & 13 & 41 & 72 & 0 \\ \end{array}$

STEP 5

Verify the remainder:
The remainder is $0$ , confirming that $x = 2$ is a solution of the polynomial equation.

STEP 6

Use the solution from part (a) to solve for $x$ in the equation $f(x) = 103$ :
Given $f(x) = 13x^3 + 15x^2 - 10x - 41$ , set $f(x) = 103$ :
$13x^3 + 15x^2 - 10x - 41 = 103$

STEP 7

Rearrange the equation:
$13x^3 + 15x^2 - 10x - 41 - 103 = 0$ $13x^3 + 15x^2 - 10x - 144 = 0$
Since we know $x = 2$ is a solution from part (a), the abdominal width when there are 103 eggs is:
$x = 2$
The abdominal width when there are 103 eggs is:
$\boxed{2}$

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Studdy Solution

STEP 1

1. We are given the polynomial $13x^3 + 15x^2 - 10x - 144 = 0$ .
2. We need to use synthetic division to verify that $x = 2$ is a solution.
3. We are given a function $f(x) = 13x^3 + 15x^2 - 10x - 41$ .
4. We need to find the abdominal width when there are 103 eggs, i.e., when $f(x) = 103$ .

STEP 2

1. Perform synthetic division of the polynomial $13x^3 + 15x^2 - 10x - 144$ by $x - 2$ .
2. Verify that the remainder is zero to confirm that $x = 2$ is a solution.
3. Use the result from part (a) to solve for $x$ in the equation $f(x) = 103$ .

STEP 3

Set up synthetic division for the polynomial $13x^3 + 15x^2 - 10x - 144$ with divisor $x - 2$ :
Write the coefficients: $13, 15, -10, -144$ .

STEP 4

STEP 5

Verify the remainder:
The remainder is $0$ , confirming that $x = 2$ is a solution of the polynomial equation.

STEP 6

Use the solution from part (a) to solve for $x$ in the equation $f(x) = 103$ :
Given $f(x) = 13x^3 + 15x^2 - 10x - 41$ , set $f(x) = 103$ :
$13x^3 + 15x^2 - 10x - 41 = 103$

STEP 7

Rearrange the equation:
$13x^3 + 15x^2 - 10x - 41 - 103 = 0$ $13x^3 + 15x^2 - 10x - 144 = 0$
Since we know $x = 2$ is a solution from part (a), the abdominal width when there are 103 eggs is:
$x = 2$
The abdominal width when there are 103 eggs is:
$\boxed{2}$

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Studdy Solution

STEP 1

STEP 2

STEP 3

Set up synthetic division for the polynomial 13x3+15x2−10x−144 13x^3 + 15x^2 - 10x - 144 13x3+15x2−10x−144 with divisor x−2 x - 2 x−2:Write the coefficients: 13,15,−10,−144 13, 15, -10, -144 13,15,−10,−144.

STEP 4

STEP 5

Verify the remainder:The remainder is 0 0 0, confirming that x=2 x = 2 x=2 is a solution of the polynomial equation.

STEP 6

Use the solution from part (a) to solve for x x x in the equation f(x)=103 f(x) = 103 f(x)=103:Given f(x)=13x3+15x2−10x−41 f(x) = 13x^3 + 15x^2 - 10x - 41 f(x)=13x3+15x2−10x−41, set f(x)=103 f(x) = 103 f(x)=103:13x3+15x2−10x−41=103 13x^3 + 15x^2 - 10x - 41 = 103 13x3+15x2−10x−41=103

STEP 7

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Set up synthetic division for the polynomial $13x^3 + 15x^2 - 10x - 144$ with divisor $x - 2$ :
Write the coefficients: $13, 15, -10, -144$ .

Verify the remainder:
The remainder is $0$ , confirming that $x = 2$ is a solution of the polynomial equation.

Use the solution from part (a) to solve for $x$ in the equation $f(x) = 103$ :
Given $f(x) = 13x^3 + 15x^2 - 10x - 41$ , set $f(x) = 103$ :
$13x^3 + 15x^2 - 10x - 41 = 103$