Answer :
Answer:
Yes, (x - 4) is a factor of polynomial function P(x) = x⁵ + 4x⁴ - 17x³ - 60x².
Step-by-step explanation:
The Factor Theorem states that if f(x) is a polynomial, and f(a) = 0, then (x - a) is a factor of f(x).
Therefore, according to the Factor Theorem, if (x - 4) is a factor of P(x) then P(4) = 0.
To determine if (x - 4) is a factor of P(x), substitute x = 4 into the function and solve.
[tex]\begin{aligned}x=4 \implies P(4)&=(4)^5+4(4)^4-17(4)^3-60(4)^2\\&=1024+4(256)-17(64)-60(16)\\&=1024+1024-1088-960\\&=2048-1088-960\\&=960-960\\&=0\end{aligned}[/tex]
As P(4) = 0, this confirms that (x - 4) is a factor of polynomial function P(x) = x⁵ + 4x⁴ - 17x³ - 60x².
[tex]\hrulefill[/tex]
Additional comments
The fully factored form of the given polynomial function is:
[tex]P(x)=x^2\left(x+3\right)\left(x-4\right)\left(x+5\right)[/tex]
