Solved on Nov 01, 2023

Expand the equation $f(x)=(3x+2)^3(x+1)$ . What is the first term and the constant term?

STEP 1

Assumptions1. The function is $f(x)=(3x+)^{3}(x+1)$ . We are asked to find the first term and the constant term when this function is expanded.

STEP 2

To find the first term of the expanded function, we need to consider the highest power of $x$ in the function. In this case, the highest power of $x$ is $4$ (from x^ in (x+2)^ and $x^1$ in $(x+1)$ ).

STEP 3

We can find the coefficient of the first term by multiplying the coefficients of $x$ in each term. The coefficient of $x$ in $(3x+2)^3$ is $3^3=27$ and the coefficient of $x$ in $(x+1)$ is $1$ .

STEP 4

Multiply the coefficients together to get the coefficient of the first term.
$First\, term\, coefficient =27 \times1$

STEP 5

Calculate the first term coefficient.
$First\, term\, coefficient =27 \times1 =27$

STEP 6

The first term of the expanded function is the coefficient of the first term times $x$ to the power of the highest power.
$First\, term = First\, term\, coefficient \times x^{4}$

STEP 7

Substitute the first term coefficient into the equation to find the first term.
$First\, term =27 \times x^{4}$

STEP 8

The first term of the expanded function is $27x^4$ .

STEP 9

To find the constant term of the expanded function, we need to consider the terms that do not contain $x$ . In this case, the constant term in $(3x+2)^3$ is $2^3=8$ and the constant term in $(x+)$ is $$.

STEP 10

We can find the constant term by multiplying the constant terms together.
$Constant\, term =8 \times$

STEP 11

Calculate the constant term.
$Constant\, term =8 \times =8$ The first term of the expanded function is $27x^4$ and the constant term is $8$ .

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Expand the equation $f(x)=(3x+2)^3(x+1)$ . What is the first term and the constant term?

STEP 1

Assumptions1. The function is $f(x)=(3x+)^{3}(x+1)$ . We are asked to find the first term and the constant term when this function is expanded.

STEP 2

To find the first term of the expanded function, we need to consider the highest power of $x$ in the function. In this case, the highest power of $x$ is $4$ (from x^ in (x+2)^ and $x^1$ in $(x+1)$ ).

STEP 3

We can find the coefficient of the first term by multiplying the coefficients of $x$ in each term. The coefficient of $x$ in $(3x+2)^3$ is $3^3=27$ and the coefficient of $x$ in $(x+1)$ is $1$ .

STEP 4

Multiply the coefficients together to get the coefficient of the first term.
$First\, term\, coefficient =27 \times1$

STEP 5

Calculate the first term coefficient.
$First\, term\, coefficient =27 \times1 =27$

STEP 6

The first term of the expanded function is the coefficient of the first term times $x$ to the power of the highest power.
$First\, term = First\, term\, coefficient \times x^{4}$

STEP 7

Substitute the first term coefficient into the equation to find the first term.
$First\, term =27 \times x^{4}$

STEP 8

The first term of the expanded function is $27x^4$ .

STEP 9

To find the constant term of the expanded function, we need to consider the terms that do not contain $x$ . In this case, the constant term in $(3x+2)^3$ is $2^3=8$ and the constant term in $(x+)$ is $$.

STEP 10

We can find the constant term by multiplying the constant terms together.
$Constant\, term =8 \times$

STEP 11

Calculate the constant term.
$Constant\, term =8 \times =8$ The first term of the expanded function is $27x^4$ and the constant term is $8$ .

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Expand the equation f(x)=(3x+2)3(x+1)f(x)=(3x+2)^3(x+1)f(x)=(3x+2)3(x+1). What is the first term and the constant term?

STEP 1

Assumptions1. The function is f(x)=(3x+)3(x+1)f(x)=(3x+)^{3}(x+1)f(x)=(3x+)3(x+1). We are asked to find the first term and the constant term when this function is expanded.

STEP 2

To find the first term of the expanded function, we need to consider the highest power of xxx in the function. In this case, the highest power of xxx is 444 (from x^ in (x+2)^ and x1x^1x1 in (x+1)(x+1)(x+1)).

STEP 3

We can find the coefficient of the first term by multiplying the coefficients of xxx in each term. The coefficient of xxx in (3x+2)3(3x+2)^3(3x+2)3 is 33=273^3=2733=27 and the coefficient of xxx in (x+1)(x+1)(x+1) is 111.

STEP 4

Multiply the coefficients together to get the coefficient of the first term.First term coefficient=27×1First\, term\, coefficient =27 \times1Firsttermcoefficient=27×1

STEP 5

Calculate the first term coefficient.First term coefficient=27×1=27First\, term\, coefficient =27 \times1 =27Firsttermcoefficient=27×1=27

STEP 6

The first term of the expanded function is the coefficient of the first term times xxx to the power of the highest power.First term=First term coefficient×x4First\, term = First\, term\, coefficient \times x^{4}Firstterm=Firsttermcoefficient×x4

STEP 7

Substitute the first term coefficient into the equation to find the first term.First term=27×x4First\, term =27 \times x^{4}Firstterm=27×x4

STEP 8

The first term of the expanded function is 27x427x^427x4.

STEP 9

To find the constant term of the expanded function, we need to consider the terms that do not contain xxx. In this case, the constant term in (3x+2)3(3x+2)^3(3x+2)3 is 23=82^3=823=8 and the constant term in (x+)(x+)(x+) is $$.

STEP 10

We can find the constant term by multiplying the constant terms together.Constant term=8×Constant\, term =8 \timesConstantterm=8×

STEP 11

Calculate the constant term.Constant term=8×=8Constant\, term =8 \times =8Constantterm=8×=8The first term of the expanded function is 27x427x^427x4 and the constant term is 888.

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Expand the equation $f(x)=(3x+2)^3(x+1)$ . What is the first term and the constant term?

Assumptions1. The function is $f(x)=(3x+)^{3}(x+1)$ . We are asked to find the first term and the constant term when this function is expanded.

To find the first term of the expanded function, we need to consider the highest power of $x$ in the function. In this case, the highest power of $x$ is $4$ (from x^ in (x+2)^ and $x^1$ in $(x+1)$ ).

We can find the coefficient of the first term by multiplying the coefficients of $x$ in each term. The coefficient of $x$ in $(3x+2)^3$ is $3^3=27$ and the coefficient of $x$ in $(x+1)$ is $1$ .

Multiply the coefficients together to get the coefficient of the first term.
$First\, term\, coefficient =27 \times1$

Calculate the first term coefficient.
$First\, term\, coefficient =27 \times1 =27$

The first term of the expanded function is the coefficient of the first term times $x$ to the power of the highest power.
$First\, term = First\, term\, coefficient \times x^{4}$

Substitute the first term coefficient into the equation to find the first term.
$First\, term =27 \times x^{4}$

The first term of the expanded function is $27x^4$ .

To find the constant term of the expanded function, we need to consider the terms that do not contain $x$ . In this case, the constant term in $(3x+2)^3$ is $2^3=8$ and the constant term in $(x+)$ is $$.

We can find the constant term by multiplying the constant terms together.
$Constant\, term =8 \times$

Calculate the constant term.
$Constant\, term =8 \times =8$ The first term of the expanded function is $27x^4$ and the constant term is $8$ .