Solved on Dec 01, 2023

Find values of $m$ and $n$ so that the polynomial $3m-63x^9 + (7-42n)x + 15$ is constant.

STEP 1

Assumptions
1. The given expression is $(3 m-63) x^{9}+(7-42 n) x+15$
2. We need to find $m$ and $n$ such that the given expression becomes a constant polynomial.
3. A constant polynomial is a polynomial of degree 0, meaning it has no variable terms.

STEP 2

For the given expression to be a constant polynomial, all the terms involving $x$ must be zero. This gives us two equations:
1. The coefficient of $x^{9}$ must be zero, i.e., $3m - 63 = 0$
2. The coefficient of $x$ must be zero, i.e., $7 - 42n = 0$

STEP 3

Let's solve the first equation $3m - 63 = 0$ for $m$ .
$3m = 63$

STEP 4

Divide both sides of the equation by 3 to solve for $m$ .
$m = \frac{63}{3}$

STEP 5

Calculate the value of $m$ .
$m = \frac{63}{3} = 21$

STEP 6

Now, let's solve the second equation $7 - 42n = 0$ for $n$ .
$42n = 7$

STEP 7

Divide both sides of the equation by 42 to solve for $n$ .
$n = \frac{7}{42}$

STEP 8

Calculate the value of $n$ .
$n = \frac{7}{42} = \frac{1}{6}$
So, the values of $m$ and $n$ that make the given expression a constant polynomial are $m = 21$ and $n = \frac{1}{6}$ .

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Find values of $m$ and $n$ so that the polynomial $3m-63x^9 + (7-42n)x + 15$ is constant.

STEP 1

Assumptions
1. The given expression is $(3 m-63) x^{9}+(7-42 n) x+15$
2. We need to find $m$ and $n$ such that the given expression becomes a constant polynomial.
3. A constant polynomial is a polynomial of degree 0, meaning it has no variable terms.

STEP 2

For the given expression to be a constant polynomial, all the terms involving $x$ must be zero. This gives us two equations:
1. The coefficient of $x^{9}$ must be zero, i.e., $3m - 63 = 0$
2. The coefficient of $x$ must be zero, i.e., $7 - 42n = 0$

STEP 3

Let's solve the first equation $3m - 63 = 0$ for $m$ .
$3m = 63$

STEP 4

Divide both sides of the equation by 3 to solve for $m$ .
$m = \frac{63}{3}$

STEP 5

Calculate the value of $m$ .
$m = \frac{63}{3} = 21$

STEP 6

Now, let's solve the second equation $7 - 42n = 0$ for $n$ .
$42n = 7$

STEP 7

Divide both sides of the equation by 42 to solve for $n$ .
$n = \frac{7}{42}$

STEP 8

Calculate the value of $n$ .
$n = \frac{7}{42} = \frac{1}{6}$
So, the values of $m$ and $n$ that make the given expression a constant polynomial are $m = 21$ and $n = \frac{1}{6}$ .

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Find values of mmm and nnn so that the polynomial 3m−63x9+(7−42n)x+153m-63x^9 + (7-42n)x + 153m−63x9+(7−42n)x+15 is constant.

STEP 1

Assumptions1. The given expression is (3m−63)x9+(7−42n)x+15(3 m-63) x^{9}+(7-42 n) x+15(3m−63)x9+(7−42n)x+152. We need to find mmm and nnn such that the given expression becomes a constant polynomial.3. A constant polynomial is a polynomial of degree 0, meaning it has no variable terms.

STEP 2

For the given expression to be a constant polynomial, all the terms involving xxx must be zero. This gives us two equations:1. The coefficient of x9x^{9}x9 must be zero, i.e., 3m−63=03m - 63 = 03m−63=02. The coefficient of xxx must be zero, i.e., 7−42n=07 - 42n = 07−42n=0

STEP 3

Let's solve the first equation 3m−63=03m - 63 = 03m−63=0 for mmm.3m=633m = 633m=63

STEP 4

Divide both sides of the equation by 3 to solve for mmm.m=633m = \frac{63}{3}m=363​

STEP 5

Calculate the value of mmm.m=633=21m = \frac{63}{3} = 21m=363​=21

STEP 6

Now, let's solve the second equation 7−42n=07 - 42n = 07−42n=0 for nnn.42n=742n = 742n=7

STEP 7

Divide both sides of the equation by 42 to solve for nnn.n=742n = \frac{7}{42}n=427​

STEP 8

Calculate the value of nnn.n=742=16n = \frac{7}{42} = \frac{1}{6}n=427​=61​So, the values of mmm and nnn that make the given expression a constant polynomial are m=21m = 21m=21 and n=16n = \frac{1}{6}n=61​.

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Find values of $m$ and $n$ so that the polynomial $3m-63x^9 + (7-42n)x + 15$ is constant.

Assumptions
1. The given expression is $(3 m-63) x^{9}+(7-42 n) x+15$
2. We need to find $m$ and $n$ such that the given expression becomes a constant polynomial.
3. A constant polynomial is a polynomial of degree 0, meaning it has no variable terms.

For the given expression to be a constant polynomial, all the terms involving $x$ must be zero. This gives us two equations:
1. The coefficient of $x^{9}$ must be zero, i.e., $3m - 63 = 0$
2. The coefficient of $x$ must be zero, i.e., $7 - 42n = 0$

Let's solve the first equation $3m - 63 = 0$ for $m$ .
$3m = 63$

Divide both sides of the equation by 3 to solve for $m$ .
$m = \frac{63}{3}$

Calculate the value of $m$ .
$m = \frac{63}{3} = 21$

Now, let's solve the second equation $7 - 42n = 0$ for $n$ .
$42n = 7$

Divide both sides of the equation by 42 to solve for $n$ .
$n = \frac{7}{42}$

Calculate the value of $n$ .
$n = \frac{7}{42} = \frac{1}{6}$
So, the values of $m$ and $n$ that make the given expression a constant polynomial are $m = 21$ and $n = \frac{1}{6}$ .