Property 5
If \(P(x)\) and \(Q(x)\) are polynomials
\(d[P(x)]=m \qquad d[Q(x)]=n\)
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\(d[P(x)\cdot Q(x)]=m+n\)
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If \(\samefrac{P(x)}{Q(x)}\) is polynomial\(d\!\biggl[\samefrac{P(x)}{Q(x)}\biggr]=m-n\)
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\(d[P(x)\mp Q(x)]=\max\{m,n\}\)\((m\neq n)\)
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\(k\in \mathbb{R}\)\(d[k\cdot P(x)]=m\)\(d[P(kx)]=m\)
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\(k\in \mathbb{R}\)\(d[P(x^k)]=k\cdot m\)\(d[P^k(x)]=k\cdot m\)
1.
\(P(x)=4x^6-3x^2\)
\(\Rightarrow d[P(x)]=?\)
2.
\(P(x)=(x^2-1)^3\)
\(\Rightarrow d[P(x)]=?\)
3.
\(P(x)=x^{n+3}-2x+3\)
\(d[P(x)]=6\)
\(\Rightarrow n=?\)
4.
\(P(x)=(x^2-1)^3\cdot x^3\)
\(\Rightarrow d[P(x)]=?\)
5.
\(P(x)=(x^2+1)^4(x-1)^3\)
\(\Rightarrow d[P(x)]=?\)
6.
\(d[P(x)]=4\)
\(\Rightarrow d[P(x^2)]=?\)
7.
\(d[P(x)]=6\)
\(d[Q(x)]=4\)
\(\Rightarrow d[P(x)\cdot Q(x)]=?\)
8.
\(d[P(x)]=8\)
\(d[Q(x)]=3\)
\(\Rightarrow d\!\left[\samefrac{P(x)}{Q(x)}\right]=?\)
9.
\(d[P^2(x)\cdot Q(x)]=8\)
\(d\!\left[\samefrac{P(x)}{Q(x)}\right]=1\)
\(\Rightarrow d[P(x)]=?\)
10.
\(d[P^3(x)\cdot Q^2(x)]=16\)
\(d\!\left[\samefrac{P(x)}{Q(x)}\right]=2\)
\(\Rightarrow d[P(x+2)]=?\)
11.
\(d[P(x)\cdot Q(3x)]=7\)
\(d[P(x)\cdot Q(x^2)]=9\)
\(\Rightarrow d[P(3x)]=?\)