If 27x 3 – 64y 3 = (Ax + By) (Cx 2 – Dy 2 + 12xy), then the value of 4A + B + 3C + 2D is:
3
This problem involves factoring a difference of cubes and then comparing the result to a given polynomial expression to find the values of unknown coefficients.
Understanding the Difference of Cubes Formula
The expression \(27x^3 - 64y^3\) is in the form of a difference of cubes, which is \(a^3 - b^3\). The standard formula for factoring the difference of two cubes is:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Factoring \(27x^3 - 64y^3\)
We need to identify 'a' and 'b' in the expression \(27x^3 - 64y^3\).
- For the first term, \(27x^3 = (3x)^3\). So, \(a = 3x\).
- For the second term, \(64y^3 = (4y)^3\). So, \(b = 4y\).
Now, applying the difference of cubes formula:
\[ 27x^3 - 64y^3 = (3x)^3 - (4y)^3 \]
\[ = (3x - 4y)((3x)^2 + (3x)(4y) + (4y)^2) \]
\[ = (3x - 4y)(9x^2 + 12xy + 16y^2) \]
Comparing Factors to Find Coefficients
The problem gives us the equation:
\[ 27x^3 - 64y^3 = (Ax + By)(Cx^2 - Dy^2 + 12xy) \]
We have factored the left side as \((3x - 4y)(9x^2 + 12xy + 16y^2)\). So, we can write:
\[ (3x - 4y)(9x^2 + 12xy + 16y^2) = (Ax + By)(Cx^2 - Dy^2 + 12xy) \]
By comparing the corresponding factors on both sides of the equation, we can determine the values of A, B, C, and D.
Comparing the First Factor: \((Ax + By)\) and \((3x - 4y)\)
- Comparing the coefficients of \(x\): \(A = 3\)
- Comparing the coefficients of \(y\): \(B = -4\)
Comparing the Second Factor: \((Cx^2 - Dy^2 + 12xy)\) and \((9x^2 + 12xy + 16y^2)\)
- Comparing the coefficients of \(x^2\): \(C = 9\)
- Comparing the coefficients of \(y^2\): \(-D = 16\), which means \(D = -16\)
- Comparing the coefficients of \(xy\): \(12 = 12\) (This matches and confirms our factoring is consistent)
So, we have found the values of the coefficients:
| Coefficient | Value |
|---|---|
| A | 3 |
| B | -4 |
| C | 9 |
| D | -16 |
Calculating \(4A + B + 3C + 2D\)
Now we need to find the value of the expression \(4A + B + 3C + 2D\) using the values we found for A, B, C, and D.
Substitute the values:
\[ 4A + B + 3C + 2D = 4(3) + (-4) + 3(9) + 2(-16) \]
Perform the multiplications:
\[ = 12 - 4 + 27 - 32 \]
Perform the additions and subtractions from left to right:
\[ = (12 - 4) + 27 - 32 \]
\[ = 8 + 27 - 32 \]
\[ = 35 - 32 \]
\[ = 3 \]
Thus, the value of \(4A + B + 3C + 2D\) is 3.
Revision Table: Algebra Factoring and Coefficients
| Concept | Description | Formula/Example |
|---|---|---|
| Difference of Cubes | A binomial where one perfect cube is subtracted from another. | \(a^3 - b^3\) |
| Factoring Difference of Cubes | Breaking down a difference of cubes into a binomial and a trinomial factor. | \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\) |
| Comparing Coefficients | Equating the coefficients of like terms in two equal polynomial expressions to find unknown values. | If \(Px + Q = 5x + 7\), then \(P=5\) and \(Q=7\). |
Additional Information: Sum of Cubes
Related to the difference of cubes is the sum of cubes. The formula for factoring the sum of two cubes is slightly different:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Notice the signs in the factored form. The binomial factor has the same sign as the sum/difference, and the trinomial factor has the opposite sign for the 'ab' term and a positive sign for the \(b^2\) term.
Example: Factoring \(8x^3 + 125y^3\)
- \(a^3 = 8x^3 \Rightarrow a = 2x\)
- \(b^3 = 125y^3 \Rightarrow b = 5y\)
Using the formula:
\[ 8x^3 + 125y^3 = (2x + 5y)((2x)^2 - (2x)(5y) + (5y)^2) \]
\[ = (2x + 5y)(4x^2 - 10xy + 25y^2) \]
Understanding both difference and sum of cubes factoring is important for algebra problems involving cubic expressions.
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