Question Find the degree 4 polynomial $P(x)$ with roots at $x=4$ (multiplicity 2), $x=0$ (multiplicity 1), and $x=-2$ (multiplicity 1), passing through $(1,-135)$ .

Studdy Solution

STEP 1

Assumptions
1. The polynomial $P(x)$ is of degree $4$ .
2. $P(x)$ has a root of multiplicity $2$ at $x=4$ .
3. $P(x)$ has roots of multiplicity $1$ at $x=0$ and $x=-2$ .
4. $P(x)$ passes through the point $(1, -135)$ .

STEP 2

The general form of a polynomial with the given roots can be written as:
$P(x) = a(x - r_1)^{n_1}(x - r_2)^{n_2}...(x - r_k)^{n_k}$
where $a$ is a leading coefficient, $r_i$ are the roots, and $n_i$ are their multiplicities.

STEP 3

Since we know the roots and their multiplicities, we can write $P(x)$ as:
$P(x) = a(x - 4)^2(x - 0)(x + 2)$

STEP 4

Expand the polynomial to find a general expression for $P(x)$ :
$P(x) = a(x^2 - 8x + 16)(x)(x + 2)$

STEP 5

Continue expanding the polynomial:
$P(x) = a(x^3 - 8x^2 + 16x)(x + 2)$

STEP 6

Finish expanding the polynomial:
$P(x) = a(x^4 - 8x^3 + 16x^2 + 2x^3 - 16x^2 + 32x)$

STEP 7

Combine like terms:
$P(x) = a(x^4 - 6x^3 + 48x)$

STEP 8

Now we use the point $(1, -135)$ to find the leading coefficient $a$ . We plug in $x = 1$ and $P(x) = -135$ :
$-135 = a(1^4 - 6 \cdot 1^3 + 48 \cdot 1)$

STEP 9

Simplify the equation:
$-135 = a(1 - 6 + 48)$

STEP 10

Further simplify the equation:
$-135 = a(43)$

STEP 11

Solve for $a$ :
$a = \frac{-135}{43}$

STEP 12

Now that we have the value of $a$ , we can write the final formula for $P(x)$ :
$P(x) = \frac{-135}{43}(x^4 - 6x^3 + 48x)$
This is the formula for the polynomial $P(x)$ that satisfies all the given conditions.

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Question Find the degree 4 polynomial $P(x)$ with roots at $x=4$ (multiplicity 2), $x=0$ (multiplicity 1), and $x=-2$ (multiplicity 1), passing through $(1,-135)$ .

Studdy Solution

STEP 1

Assumptions
1. The polynomial $P(x)$ is of degree $4$ .
2. $P(x)$ has a root of multiplicity $2$ at $x=4$ .
3. $P(x)$ has roots of multiplicity $1$ at $x=0$ and $x=-2$ .
4. $P(x)$ passes through the point $(1, -135)$ .

STEP 2

The general form of a polynomial with the given roots can be written as:
$P(x) = a(x - r_1)^{n_1}(x - r_2)^{n_2}...(x - r_k)^{n_k}$
where $a$ is a leading coefficient, $r_i$ are the roots, and $n_i$ are their multiplicities.

STEP 3

Since we know the roots and their multiplicities, we can write $P(x)$ as:
$P(x) = a(x - 4)^2(x - 0)(x + 2)$

STEP 4

Expand the polynomial to find a general expression for $P(x)$ :
$P(x) = a(x^2 - 8x + 16)(x)(x + 2)$

STEP 5

Continue expanding the polynomial:
$P(x) = a(x^3 - 8x^2 + 16x)(x + 2)$

STEP 6

Finish expanding the polynomial:
$P(x) = a(x^4 - 8x^3 + 16x^2 + 2x^3 - 16x^2 + 32x)$

STEP 7

Combine like terms:
$P(x) = a(x^4 - 6x^3 + 48x)$

STEP 8

Now we use the point $(1, -135)$ to find the leading coefficient $a$ . We plug in $x = 1$ and $P(x) = -135$ :
$-135 = a(1^4 - 6 \cdot 1^3 + 48 \cdot 1)$

STEP 9

Simplify the equation:
$-135 = a(1 - 6 + 48)$

STEP 10

Further simplify the equation:
$-135 = a(43)$

STEP 11

Solve for $a$ :
$a = \frac{-135}{43}$

STEP 12

Now that we have the value of $a$ , we can write the final formula for $P(x)$ :
$P(x) = \frac{-135}{43}(x^4 - 6x^3 + 48x)$
This is the formula for the polynomial $P(x)$ that satisfies all the given conditions.

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Question Find the degree 4 polynomial P(x)P(x)P(x) with roots at x=4x=4x=4 (multiplicity 2), x=0x=0x=0 (multiplicity 1), and x=−2x=-2x=−2 (multiplicity 1), passing through (1,−135)(1,-135)(1,−135).

Studdy Solution

STEP 1

Assumptions1. The polynomial P(x)P(x)P(x) is of degree 444.2. P(x)P(x)P(x) has a root of multiplicity 222 at x=4x=4x=4.3. P(x)P(x)P(x) has roots of multiplicity 111 at x=0x=0x=0 and x=−2x=-2x=−2.4. P(x)P(x)P(x) passes through the point (1,−135)(1, -135)(1,−135).

STEP 2

STEP 3

Since we know the roots and their multiplicities, we can write P(x)P(x)P(x) as:P(x)=a(x−4)2(x−0)(x+2)P(x) = a(x - 4)^2(x - 0)(x + 2)P(x)=a(x−4)2(x−0)(x+2)

STEP 4

Expand the polynomial to find a general expression for P(x)P(x)P(x):P(x)=a(x2−8x+16)(x)(x+2)P(x) = a(x^2 - 8x + 16)(x)(x + 2)P(x)=a(x2−8x+16)(x)(x+2)

STEP 5

Continue expanding the polynomial:P(x)=a(x3−8x2+16x)(x+2)P(x) = a(x^3 - 8x^2 + 16x)(x + 2)P(x)=a(x3−8x2+16x)(x+2)

STEP 6

Finish expanding the polynomial:P(x)=a(x4−8x3+16x2+2x3−16x2+32x)P(x) = a(x^4 - 8x^3 + 16x^2 + 2x^3 - 16x^2 + 32x)P(x)=a(x4−8x3+16x2+2x3−16x2+32x)

STEP 7

Combine like terms:P(x)=a(x4−6x3+48x)P(x) = a(x^4 - 6x^3 + 48x)P(x)=a(x4−6x3+48x)

STEP 8

Now we use the point (1,−135)(1, -135)(1,−135) to find the leading coefficient aaa. We plug in x=1x = 1x=1 and P(x)=−135P(x) = -135P(x)=−135:−135=a(14−6⋅13+48⋅1)-135 = a(1^4 - 6 \cdot 1^3 + 48 \cdot 1)−135=a(14−6⋅13+48⋅1)

STEP 9

Simplify the equation:−135=a(1−6+48)-135 = a(1 - 6 + 48)−135=a(1−6+48)

STEP 10

Further simplify the equation:−135=a(43)-135 = a(43)−135=a(43)

STEP 11

Solve for aaa:a=−13543a = \frac{-135}{43}a=43−135​

STEP 12

Now that we have the value of aaa, we can write the final formula for P(x)P(x)P(x):P(x)=−13543(x4−6x3+48x)P(x) = \frac{-135}{43}(x^4 - 6x^3 + 48x)P(x)=43−135​(x4−6x3+48x)This is the formula for the polynomial P(x)P(x)P(x) that satisfies all the given conditions.

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Question Find the degree 4 polynomial $P(x)$ with roots at $x=4$ (multiplicity 2), $x=0$ (multiplicity 1), and $x=-2$ (multiplicity 1), passing through $(1,-135)$ .

Assumptions
1. The polynomial $P(x)$ is of degree $4$ .
2. $P(x)$ has a root of multiplicity $2$ at $x=4$ .
3. $P(x)$ has roots of multiplicity $1$ at $x=0$ and $x=-2$ .
4. $P(x)$ passes through the point $(1, -135)$ .

Since we know the roots and their multiplicities, we can write $P(x)$ as:
$P(x) = a(x - 4)^2(x - 0)(x + 2)$

Expand the polynomial to find a general expression for $P(x)$ :
$P(x) = a(x^2 - 8x + 16)(x)(x + 2)$

Continue expanding the polynomial:
$P(x) = a(x^3 - 8x^2 + 16x)(x + 2)$

Finish expanding the polynomial:
$P(x) = a(x^4 - 8x^3 + 16x^2 + 2x^3 - 16x^2 + 32x)$

Combine like terms:
$P(x) = a(x^4 - 6x^3 + 48x)$

Now we use the point $(1, -135)$ to find the leading coefficient $a$ . We plug in $x = 1$ and $P(x) = -135$ :
$-135 = a(1^4 - 6 \cdot 1^3 + 48 \cdot 1)$

Simplify the equation:
$-135 = a(1 - 6 + 48)$

Further simplify the equation:
$-135 = a(43)$

Solve for $a$ :
$a = \frac{-135}{43}$

Now that we have the value of $a$ , we can write the final formula for $P(x)$ :
$P(x) = \frac{-135}{43}(x^4 - 6x^3 + 48x)$
This is the formula for the polynomial $P(x)$ that satisfies all the given conditions.