Reasoning vs Computation Math Example 4

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Example 4

medium
By reasoning (not expanding), determine the degree of the polynomial (x3+2)4(x2−1)5(x^3+2)^4(x^2-1)^5.

Solution

  1. 1
    The degree of a product is the sum of the degrees of the factors.
  2. 2
    Degree of (x3+2)4(x^3+2)^4: the leading term is x3×4=x12x^{3 \times 4} = x^{12}, so degree 12.
  3. 3
    Degree of (x2−1)5(x^2-1)^5: the leading term is x2×5=x10x^{2 \times 5} = x^{10}, so degree 10.
  4. 4
    Total degree: 12+10=2212 + 10 = 22.

Answer

deg⁡[(x3+2)4(x2−1)5]=22\deg\left[(x^3+2)^4(x^2-1)^5\right] = 22
Reasoning about degrees using the multiplicativity property avoids expanding a degree-22 polynomial. Structural properties of degrees allow us to bypass the computation entirely.

About Reasoning vs Computation

Reasoning is the process of understanding why a mathematical fact is true and how ideas connect, while computation is the mechanical process of calculating an answer — both are essential but serve different purposes.

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Use reasoning (not direct computation) to find the units digit of [formula].

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