Question

which polynomial function f(x) has a leading coefficient of 1, roots -4, 2, and 9 with multiplicity 1, and root -5 with multiplicity 3?\\(\circ\\) \\(f(x) = 3(x + 5)(x + 4)(x - 2)(x - 9)\\)\\(\circ\\) \\(f(x) = 3(x - 5)(x - 4)(x + 2)(x + 9)\\)\\(\circ\\) \\(f(x) = (x + 5)(x + 5)(x + 5)(x + 4)(x - 2)(x - 9)\\)\\(\circ\\) \\(f(x) = (x - 5)(x - 5)(x - 5)(x - 4)(x + 2)(x + 9)\\)

Explanation:

Step1: Recall Polynomial Root Form

If a polynomial has a root \( r \) with multiplicity \( m \), then \( (x - r)^m \) is a factor. Leading coefficient is the coefficient of the highest - degree term.

Step2: Analyze Roots and Multiplicities

• Root \(- 4\) with multiplicity \( 1\): The factor is \( (x+4)=(x - (-4))^1\)
• Root \( 2\) with multiplicity \( 1\): The factor is \( (x - 2)=(x - 2)^1\)
• Root \( 9\) with multiplicity \( 1\): The factor is \( (x - 9)=(x - 9)^1\)
• Root \(-5\) with multiplicity \( 3\): The factor is \( (x + 5)^3=(x-(-5))^3=(x + 5)(x + 5)(x + 5)\)
• Leading coefficient is \( 1\), so there is no coefficient other than \( 1\) in front of the product of factors.

Step3: Construct the Polynomial

Combining the factors, the polynomial \( f(x)=(x + 5)(x + 5)(x + 5)(x + 4)(x - 2)(x - 9)\)

Step4: Eliminate Other Options

• Option 1: \( f(x)=3(x + 5)(x + 4)(x - 2)(x - 9)\) has a leading coefficient of \( 3\) (not \( 1\)) and does not account for the multiplicity of \(-5\) (multiplicity \( 1\) instead of \( 3\)), so eliminate.
• Option 2: \( f(x)=3(x - 5)(x - 4)(x + 2)(x + 9)\) has a leading coefficient of \( 3\) (not \( 1\)) and the roots are incorrect (\(5,4,-2,-9\) instead of \(-4,2,9,-5\)), so eliminate.
• Option 4: \( f(x)=(x - 5)(x - 5)(x - 5)(x - 4)(x + 2)(x + 9)\) has incorrect roots (\(5,4,-2,-9\) instead of \(-4,2,9,-5\)), so eliminate.

Answer:

\( f(x)=(x + 5)(x + 5)(x + 5)(x + 4)(x - 2)(x - 9) \) (the third option)