Solved on Mar 07, 2024

Convert cubic function to standard form. Find $a, b, c, d$ where $y = ax^3 + bx^2 + cx + d$ .

STEP 1

Assumptions
1. We are given a cubic function in the form $y=-2(x+4)^{3}-5$ .
2. We need to convert it to standard form, which is $y=ax^{3}+bx^{2}+cx+d$ .
3. We will expand the given cubic function using algebraic identities and simplification.

STEP 2

The first step is to expand the cubic term $(-2(x+4)^{3})$ . We will use the binomial expansion for the cube of a binomial.
$(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

STEP 3

Apply the binomial expansion to $(x+4)^3$ .
$(x+4)^3 = x^3 + 3(x^2)(4) + 3(x)(4^2) + 4^3$

STEP 4

Simplify the expansion by calculating the constants.
$(x+4)^3 = x^3 + 3 \cdot 4 \cdot x^2 + 3 \cdot 16 \cdot x + 64$

STEP 5

Now, we have the expanded form of $(x+4)^3$ .
$(x+4)^3 = x^3 + 12x^2 + 48x + 64$

STEP 6

Next, we multiply the expanded form by the coefficient $-2$ that is outside the cubic term.
$-2(x+4)^3 = -2(x^3 + 12x^2 + 48x + 64)$

STEP 7

Distribute the $-2$ across all terms in the expansion.
$-2(x+4)^3 = -2x^3 - 2 \cdot 12x^2 - 2 \cdot 48x - 2 \cdot 64$

STEP 8

Simplify the multiplication to get the terms of the expanded cubic expression.
$-2(x+4)^3 = -2x^3 - 24x^2 - 96x - 128$

STEP 9

Now, we incorporate the constant term $-5$ from the original function.
$y = -2(x+4)^3 - 5$

STEP 10

Substitute the expanded form of $-2(x+4)^3$ into the equation.
$y = (-2x^3 - 24x^2 - 96x - 128) - 5$

STEP 11

Combine the constant terms $-128$ and $-5$ .
$y = -2x^3 - 24x^2 - 96x - 128 - 5$

STEP 12

Simplify the constant terms.
$y = -2x^3 - 24x^2 - 96x - 133$

STEP 13

We have now converted the given function to standard form. We can identify the coefficients $a, b, c$ , and $d$ .
$a = -2$ $b = -24$ $c = -96$ $d = -133$
The cubic function in standard form is:
$y = -2x^3 - 24x^2 - 96x - 133$
Therefore, the values for $a, b, c$ , and $d$ are:
$a = -2$ $b = -24$ $c = -96$ $d = -133$

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Convert cubic function to standard form. Find $a, b, c, d$ where $y = ax^3 + bx^2 + cx + d$ .

STEP 1

Assumptions
1. We are given a cubic function in the form $y=-2(x+4)^{3}-5$ .
2. We need to convert it to standard form, which is $y=ax^{3}+bx^{2}+cx+d$ .
3. We will expand the given cubic function using algebraic identities and simplification.

STEP 2

The first step is to expand the cubic term $(-2(x+4)^{3})$ . We will use the binomial expansion for the cube of a binomial.
$(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

STEP 3

Apply the binomial expansion to $(x+4)^3$ .
$(x+4)^3 = x^3 + 3(x^2)(4) + 3(x)(4^2) + 4^3$

STEP 4

Simplify the expansion by calculating the constants.
$(x+4)^3 = x^3 + 3 \cdot 4 \cdot x^2 + 3 \cdot 16 \cdot x + 64$

STEP 5

Now, we have the expanded form of $(x+4)^3$ .
$(x+4)^3 = x^3 + 12x^2 + 48x + 64$

STEP 6

Next, we multiply the expanded form by the coefficient $-2$ that is outside the cubic term.
$-2(x+4)^3 = -2(x^3 + 12x^2 + 48x + 64)$

STEP 7

Distribute the $-2$ across all terms in the expansion.
$-2(x+4)^3 = -2x^3 - 2 \cdot 12x^2 - 2 \cdot 48x - 2 \cdot 64$

STEP 8

Simplify the multiplication to get the terms of the expanded cubic expression.
$-2(x+4)^3 = -2x^3 - 24x^2 - 96x - 128$

STEP 9

Now, we incorporate the constant term $-5$ from the original function.
$y = -2(x+4)^3 - 5$

STEP 10

Substitute the expanded form of $-2(x+4)^3$ into the equation.
$y = (-2x^3 - 24x^2 - 96x - 128) - 5$

STEP 11

Combine the constant terms $-128$ and $-5$ .
$y = -2x^3 - 24x^2 - 96x - 128 - 5$

STEP 12

Simplify the constant terms.
$y = -2x^3 - 24x^2 - 96x - 133$

STEP 13

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Convert cubic function to standard form. Find a,b,c,da, b, c, da,b,c,d where y=ax3+bx2+cx+dy = ax^3 + bx^2 + cx + dy=ax3+bx2+cx+d.

STEP 1

Assumptions1. We are given a cubic function in the form y=−2(x+4)3−5y=-2(x+4)^{3}-5y=−2(x+4)3−5.2. We need to convert it to standard form, which is y=ax3+bx2+cx+dy=ax^{3}+bx^{2}+cx+dy=ax3+bx2+cx+d.3. We will expand the given cubic function using algebraic identities and simplification.

STEP 2

The first step is to expand the cubic term (−2(x+4)3)(-2(x+4)^{3})(−2(x+4)3). We will use the binomial expansion for the cube of a binomial.(a+b)3=a3+3a2b+3ab2+b3 (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 (a+b)3=a3+3a2b+3ab2+b3

STEP 3

Apply the binomial expansion to (x+4)3(x+4)^3(x+4)3.(x+4)3=x3+3(x2)(4)+3(x)(42)+43 (x+4)^3 = x^3 + 3(x^2)(4) + 3(x)(4^2) + 4^3 (x+4)3=x3+3(x2)(4)+3(x)(42)+43

STEP 4

Simplify the expansion by calculating the constants.(x+4)3=x3+3⋅4⋅x2+3⋅16⋅x+64 (x+4)^3 = x^3 + 3 \cdot 4 \cdot x^2 + 3 \cdot 16 \cdot x + 64 (x+4)3=x3+3⋅4⋅x2+3⋅16⋅x+64

STEP 5

Now, we have the expanded form of (x+4)3(x+4)^3(x+4)3.(x+4)3=x3+12x2+48x+64 (x+4)^3 = x^3 + 12x^2 + 48x + 64 (x+4)3=x3+12x2+48x+64

STEP 6

Next, we multiply the expanded form by the coefficient −2-2−2 that is outside the cubic term.−2(x+4)3=−2(x3+12x2+48x+64) -2(x+4)^3 = -2(x^3 + 12x^2 + 48x + 64) −2(x+4)3=−2(x3+12x2+48x+64)

STEP 7

Distribute the −2-2−2 across all terms in the expansion.−2(x+4)3=−2x3−2⋅12x2−2⋅48x−2⋅64 -2(x+4)^3 = -2x^3 - 2 \cdot 12x^2 - 2 \cdot 48x - 2 \cdot 64 −2(x+4)3=−2x3−2⋅12x2−2⋅48x−2⋅64

STEP 8

Simplify the multiplication to get the terms of the expanded cubic expression.−2(x+4)3=−2x3−24x2−96x−128 -2(x+4)^3 = -2x^3 - 24x^2 - 96x - 128 −2(x+4)3=−2x3−24x2−96x−128

STEP 9

Now, we incorporate the constant term −5-5−5 from the original function.y=−2(x+4)3−5 y = -2(x+4)^3 - 5 y=−2(x+4)3−5

STEP 10

Substitute the expanded form of −2(x+4)3-2(x+4)^3−2(x+4)3 into the equation.y=(−2x3−24x2−96x−128)−5 y = (-2x^3 - 24x^2 - 96x - 128) - 5 y=(−2x3−24x2−96x−128)−5

STEP 11

Combine the constant terms −128-128−128 and −5-5−5.y=−2x3−24x2−96x−128−5 y = -2x^3 - 24x^2 - 96x - 128 - 5 y=−2x3−24x2−96x−128−5

STEP 12

Simplify the constant terms.y=−2x3−24x2−96x−133 y = -2x^3 - 24x^2 - 96x - 133 y=−2x3−24x2−96x−133

STEP 13

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Convert cubic function to standard form. Find $a, b, c, d$ where $y = ax^3 + bx^2 + cx + d$ .

Assumptions
1. We are given a cubic function in the form $y=-2(x+4)^{3}-5$ .
2. We need to convert it to standard form, which is $y=ax^{3}+bx^{2}+cx+d$ .
3. We will expand the given cubic function using algebraic identities and simplification.

The first step is to expand the cubic term $(-2(x+4)^{3})$ . We will use the binomial expansion for the cube of a binomial.
$(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

Apply the binomial expansion to $(x+4)^3$ .
$(x+4)^3 = x^3 + 3(x^2)(4) + 3(x)(4^2) + 4^3$

Simplify the expansion by calculating the constants.
$(x+4)^3 = x^3 + 3 \cdot 4 \cdot x^2 + 3 \cdot 16 \cdot x + 64$

Now, we have the expanded form of $(x+4)^3$ .
$(x+4)^3 = x^3 + 12x^2 + 48x + 64$

Next, we multiply the expanded form by the coefficient $-2$ that is outside the cubic term.
$-2(x+4)^3 = -2(x^3 + 12x^2 + 48x + 64)$

Distribute the $-2$ across all terms in the expansion.
$-2(x+4)^3 = -2x^3 - 2 \cdot 12x^2 - 2 \cdot 48x - 2 \cdot 64$

Simplify the multiplication to get the terms of the expanded cubic expression.
$-2(x+4)^3 = -2x^3 - 24x^2 - 96x - 128$

Now, we incorporate the constant term $-5$ from the original function.
$y = -2(x+4)^3 - 5$

Substitute the expanded form of $-2(x+4)^3$ into the equation.
$y = (-2x^3 - 24x^2 - 96x - 128) - 5$

Combine the constant terms $-128$ and $-5$ .
$y = -2x^3 - 24x^2 - 96x - 128 - 5$

Simplify the constant terms.
$y = -2x^3 - 24x^2 - 96x - 133$