Question
7. Multiply: \( \left(3 x^{2}-4\right)\left(4 x^{2}-9 y\right) \)
Ask by Rose Gonzalez. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
Multiply the binomials to get \( 12x^{4} - 27x^{2}y - 16x^{2} + 36y \).
Solution
To multiply the binomials \( (3x^{2} - 4) \) and \( (4x^{2} - 9y) \), we'll use the distributive property (also known as the FOIL method for binomials). Here's a step-by-step breakdown:
\[
\begin{align*}
(3x^{2} - 4)(4x^{2} - 9y) &= 3x^{2} \cdot 4x^{2} + 3x^{2} \cdot (-9y) + (-4) \cdot 4x^{2} + (-4) \cdot (-9y) \\
&= 12x^{4} - 27x^{2}y - 16x^{2} + 36y
\end{align*}
\]
**Final Expanded Form:**
\[
12x^{4} - 27x^{2}y - 16x^{2} + 36y
\]
If desired, you can factor out common terms:
\[
12x^{4} - (27y + 16)x^{2} + 36y
\]
However, unless further factoring is possible or required, the expanded form above is typically sufficient.
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Simplify this solution Bonus Knowledge
To multiply the expressions \( (3x^{2} - 4)(4x^{2} - 9y) \), we can use the distributive property (also known as the FOIL method for binomials). This involves multiplying each term in the first polynomial by each term in the second polynomial. 1. First, multiply the first terms: \( 3x^{2} \cdot 4x^{2} = 12x^{4} \). 2. Next, multiply the outer terms: \( 3x^{2} \cdot -9y = -27xy \). 3. Then, multiply the inner terms: \( -4 \cdot 4x^{2} = -16x^{2} \). 4. Lastly, multiply the last terms: \( -4 \cdot -9y = 36y \). Now, combine all these results together: \[ 12x^{4} - 27xy - 16x^{2} + 36y \] So, the final result of the multiplication is: \[ 12x^{4} - 16x^{2} - 27xy + 36y \]
