Given that logm3≈0.93, logm4≈1.174, and logm7≈1.647, find logm4m .

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anarosa52505pdizws

20.12.2022

Mathematics

Among the four choices, (x + 5) is the only one that is not a linear factor of this polynomial function.

Step-by-step explanation:

Let denote some constant. A linear factor of the form (x - a) is a factor of a polynomial f(x) if and only if f(a) = 0 (that is: replacing all in the polynomial $f(x) \!$ with the constant $a\!$ would give this polynomial a value of .)

For example, in the second linear factor (x - 2) , the value of the constant is a = 2 . Verify that the value of f(2) is indeed . (In other words, replacing all in the polynomial $f(x) \!$ with the constant should give this polynomial a value of $0\!$ .)

$\begin{aligned}f(2) &= 2^3 - 5\times 2^2 - 4 \times 2 + 20 \\ &= 8 - 20 - 8 + 20 \\ &= 0 \end{aligned}$ .

Hence, (x - 2) is indeed a linear factor of polynomial f(x) .

Similarly, it could be verified that (x - 5) and (x + 2) are also linear factors of this polynomial function.

Rewrite the first linear factor (x + 5) in the form (x - a) for some constant : (x + 5) = (x - (-5)) , where a = -5 .

Calculate the value of f(5) .

$\begin{aligned}f(5) &= (-5)^3 - 5\times (-5)^2 - 4 \times (-5) + 20 \\ &= (-125) - 125 + 20 + 20 \\ &= -210\end{aligned}$ .

$f(5) \ne 0$ implies that (x - (-5)) (which is equivalent to (x + 5) ) isn't a linear factor of this polynomial function.

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