Which expression represents the product of x^3 + 2x – 1 and x^4 – x^3 + 3?
The Correct Answer and Explanation is:
To find the product of the expressions (x3+2x−1)(x4−x3+3)(x^3 + 2x – 1)(x^4 – x^3 + 3)(x3+2x−1)(x4−x3+3), apply the distributive property (also called the FOIL method in binomials, but extended here for polynomials). Multiply each term in the first polynomial by every term in the second polynomial:
Step-by-step multiplication:
1. Multiply x3x^3×3 with each term of the second polynomial: x3⋅x4=x7x3⋅(−x3)=−x6x3⋅3=3x3x^3 \cdot x^4 = x^7 \\ x^3 \cdot (-x^3) = -x^6 \\ x^3 \cdot 3 = 3x^3×3⋅x4=x7x3⋅(−x3)=−x6x3⋅3=3×3
2. Multiply 2x2x2x with each term of the second polynomial: 2x⋅x4=2x52x⋅(−x3)=−2x42x⋅3=6x2x \cdot x^4 = 2x^5 \\ 2x \cdot (-x^3) = -2x^4 \\ 2x \cdot 3 = 6x2x⋅x4=2x52x⋅(−x3)=−2x42x⋅3=6x
3. Multiply −1-1−1 with each term of the second polynomial: −1⋅x4=−x4−1⋅(−x3)=x3−1⋅3=−3-1 \cdot x^4 = -x^4 \\ -1 \cdot (-x^3) = x^3 \\ -1 \cdot 3 = -3−1⋅x4=−x4−1⋅(−x3)=x3−1⋅3=−3
Now combine all the resulting terms: x7−x6+3×3+2×5−2×4+6x−x4+x3−3x^7 – x^6 + 3x^3 + 2x^5 – 2x^4 + 6x – x^4 + x^3 – 3×7−x6+3×3+2×5−2×4+6x−x4+x3−3
Combine like terms:
- x7x^7×7 (no like term)
- −x6-x^6−x6 (no like term)
- 2x52x^52×5 (no like term)
- −2×4−x4=−3×4-2x^4 – x^4 = -3x^4−2×4−x4=−3×4
- 3×3+x3=4x33x^3 + x^3 = 4x^33×3+x3=4×3
- 6x6x6x (no like term)
- −3-3−3 (constant term)
Final expression:
x7−x6+2×5−3×4+4×3+6x−3x^7 – x^6 + 2x^5 – 3x^4 + 4x^3 + 6x – 3×7−x6+2×5−3×4+4×3+6x−3
This polynomial is the result of multiplying x3+2x−1x^3 + 2x – 1×3+2x−1 and x4−x3+3x^4 – x^3 + 3×4−x3+3. Each step relies on the principle that multiplying powers of the same base involves adding their exponents, and combining like terms involves summing their coefficients. This method is fundamental in algebraic expansion and is useful for simplifying or preparing expressions for further analysis.
