To expand the expression \( (2x^2 + x + 3)(3x^2 - 2x + 6) \), we will use the distributive property (also known as the FOIL method for two binomials). Here, we'll treat one polynomial as being multiplied by each term of the other polynomial.
Let's go through the expansion step-by-step:
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Multiply each term in the first polynomial by each term in the second polynomial:
\[ (2x^2)(3x^2) + (2x^2)(-2x) + (2x^2)(6) + (x)(3x^2) + (x)(-2x) + (x)(6) + (3)(3x^2) + (3)(-2x) + (3)(6) \]
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Calculate each term:
- \( (2x^2)(3x^2) = 6x^4 \)
- \( (2x^2)(-2x) = -4x^3 \)
- \( (2x^2)(6) = 12x^2 \)
- \( (x)(3x^2) = 3x^3 \)
- \( (x)(-2x) = -2x^2 \)
- \( (x)(6) = 6x \)
- \( (3)(3x^2) = 9x^2 \)
- \( (3)(-2x) = -6x \)
- \( (3)(6) = 18 \)
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Combine all the terms:
\[ 6x^4 + (-4x^3 + 3x^3) + (12x^2 - 2x^2 + 9x^2) + (6x - 6x) + 18 \]
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Combine like terms:
- For \(x^4\): \(6x^4\)
- For \(x^3\): \(-4x^3 + 3x^3 = -1x^3\)
- For \(x^2\): \(12x^2 - 2x^2 + 9x^2 = 19x^2\)
- For \(x\): \(6x - 6x = 0\)
- Constant: \(18\)
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Write the final polynomial in standard form:
The final expanded polynomial in standard form (highest degree to lowest) is:
\[ 6x^4 - x^3 + 19x^2 + 18 \]
Regarding your other questions:
- Standard Form: In polynomial expressions, standard form means writing the polynomial in descending order of power, which we've done above.
- Parentheses: There are no parentheses in the final standard polynomial form.
- Boxed: When a solution is "boxed," it typically refers to presenting or highlighting the final answer in a box to distinguish it from other calculations or notes, which is often done in educational materials.
If you have any further questions or need clarification, feel free to ask!
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