Determine the Possible Number of Real Roots f(x)=(x-6)^2(x+2)^2

Problem

ƒ(x)=(x−6)^2*(x+2)^2

Solution

1. Identify the degree of the polynomial by summing the exponents of the factors.

2. Determine the total number of roots using the Fundamental Theorem of Algebra, which states that a polynomial of degree n has exactly n roots (counting multiplicity).

n=2+2=4

3. Analyze the factors to find the real roots and their multiplicities.

4. Observe that the factor (x−6)^2 yields a real root at x=6 with a multiplicity of 2.

5. Observe that the factor (x+2)^2 yields a real root at x=−2 with a multiplicity of 2.

6. Conclude that since all factors are linear and squared, all 4 roots are real.

Final Answer

Number of Real Roots=4

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