Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=-4 \, h\left(x\right)^{2} - 2 \, h'\left(x\right)\hspace{3em}T(h(x))=2 \, h\left(-5\right) + 2 \, h\left(x^{2}\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=5 \, x h\left(x\right) + h'\left(x\right)\hspace{3em}T(h(x))=-h\left(x\right) h'\left(x\right) - 5 \, h\left(-2\right)\]

\(S\) is linear, and \(T\) is non-linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=-5 \, g\left(x\right)^{3} - 4 \, g\left(-4\right)\hspace{3em}T(g(x))=-4 \, g\left(x^{3}\right) - g'\left(4\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=2 \, g\left(x\right) g'\left(x\right) - 5 \, g'\left(-3\right)\hspace{3em}T(g(x))=-3 \, g\left(5\right) - 2 \, g'\left(x\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=4 \, x^{2} f\left(x\right) - 4 \, f'\left(x\right)\hspace{3em}T(f(x))=-5 \, f\left(-3\right) + 1\]

\(S\) is linear, and \(T\) is non-linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=2 \, h\left(x\right)^{2} - 3 \, h\left(x^{2}\right)\hspace{3em}T(h(x))=5 \, h\left(-4\right) + 5 \, h'\left(x\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=5 \, x^{3} h\left(x\right) + h\left(x\right)^{2}\hspace{3em}T(h(x))=-2 \, h\left(x^{2}\right) + h'\left(4\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=2 \, h\left(x\right) h'\left(x\right) - 4 \, h\left(-5\right)\hspace{3em}T(h(x))=3 \, h\left(x^{2}\right) + h'\left(3\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=3 \, g\left(x\right)^{3} + g\left(x^{2}\right)\hspace{3em}T(g(x))=4 \, x g\left(x\right) + 5 \, g\left(-2\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=5 \, x h\left(x\right) + 5 \, h'\left(x\right)\hspace{3em}T(h(x))=h\left(x\right)^{3} + 5 \, h\left(x^{2}\right)\]

\(S\) is linear, and \(T\) is non-linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=-4 \, g'\left(-2\right) + 2 \, g'\left(x\right)\hspace{3em}T(g(x))=x g\left(x\right) - 4\]

\(S\) is linear, and \(T\) is non-linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=-3 \, g\left(x\right) g'\left(x\right) - 3 \, g'\left(x\right)\hspace{3em}T(g(x))=4 \, x^{2} g\left(x\right) + 5 \, g\left(x^{2}\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=-4 \, f\left(x\right)^{3} + f\left(x\right)\hspace{3em}T(f(x))=x^{3} f\left(x\right) - f'\left(x\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=-2 \, f\left(x\right) f'\left(x\right) + 4 \, f\left(x\right)\hspace{3em}T(f(x))=5 \, f\left(x^{3}\right) + 4 \, f'\left(2\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=2 \, x^{2} h\left(x\right) - 4 \, x\hspace{3em}T(h(x))=-3 \, h'\left(-4\right) - 5 \, h'\left(x\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=3 \, g\left(x\right) + 2 \, g'\left(x\right)\hspace{3em}T(g(x))=-4 \, g\left(x\right) g'\left(x\right) + 2 \, g'\left(1\right)\]

\(S\) is linear, and \(T\) is non-linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=g\left(3\right) + 4 \, g\left(x^{2}\right)\hspace{3em}T(g(x))=-g\left(x\right) g'\left(x\right) + g'\left(x\right)\]

\(S\) is linear, and \(T\) is non-linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=2 \, h\left(x^{2}\right) + 4 \, h'\left(x\right)\hspace{3em}T(h(x))=-5 \, x - 3 \, h\left(-5\right)\]

\(S\) is linear, and \(T\) is non-linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=4 \, x^{3} g\left(x\right) + g\left(2\right)\hspace{3em}T(g(x))=5 \, x^{2} - g'\left(x\right)\]

\(S\) is linear, and \(T\) is non-linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=2 \, x^{2} g\left(x\right) - g\left(x^{2}\right)\hspace{3em}T(g(x))=-3 \, g\left(x\right) g'\left(x\right) + g'\left(x\right)\]

\(S\) is linear, and \(T\) is non-linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=-4 \, f\left(x\right) f'\left(x\right) - 3 \, f\left(x\right)\hspace{3em}T(f(x))=-f'\left(-4\right) + f'\left(x\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=4 \, x^{3} - 2 \, f'\left(3\right)\hspace{3em}T(f(x))=2 \, x f\left(x\right) + 2 \, f\left(x\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=-f\left(x\right)^{2} - 5 \, f'\left(x\right)\hspace{3em}T(f(x))=-4 \, f\left(x\right) - 5 \, f'\left(-3\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=4 \, g\left(x\right)^{3} - 2 \, g'\left(-2\right)\hspace{3em}T(g(x))=3 \, g\left(x\right) - 2 \, g'\left(x\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=4 \, g\left(2\right) - 3 \, g'\left(x\right)\hspace{3em}T(g(x))=-4 \, g\left(x\right) g'\left(x\right) - 3 \, g'\left(2\right)\]

\(S\) is linear, and \(T\) is non-linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=2 \, x^{2} h\left(x\right) + 2 \, h\left(3\right)\hspace{3em}T(h(x))=4 \, x^{3} + 4 \, h'\left(x\right)\]

\(S\) is linear, and \(T\) is non-linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=-5 \, x^{3} f\left(x\right) - 3 \, f'\left(x\right)\hspace{3em}T(f(x))=3 \, f\left(x\right)^{2} - 3 \, f'\left(1\right)\]

\(S\) is linear, and \(T\) is non-linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=-2 \, x^{3} g\left(x\right) + 4 \, g\left(x\right)^{2}\hspace{3em}T(g(x))=-2 \, g'\left(4\right) + 5 \, g'\left(x\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=-2 \, h\left(x\right)^{3} - 3 \, h\left(-3\right)\hspace{3em}T(h(x))=-3 \, h\left(x\right) - 5 \, h'\left(2\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=-2 \, x^{3} - 5 \, g'\left(-4\right)\hspace{3em}T(g(x))=5 \, g\left(x\right) - g'\left(x\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=-2 \, f\left(-5\right) - 2 \, f'\left(x\right)\hspace{3em}T(f(x))=2 \, x^{2} f\left(x\right) - 2 \, f\left(x\right)^{3}\]

\(S\) is linear, and \(T\) is non-linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=-5 \, g\left(x\right) g'\left(x\right) + g\left(x^{3}\right)\hspace{3em}T(g(x))=-2 \, g\left(2\right) + 3 \, g'\left(4\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=-5 \, f\left(1\right) + 5 \, f\left(x^{3}\right)\hspace{3em}T(f(x))=4 \, x f\left(x\right) + f\left(x\right) f'\left(x\right)\]

\(S\) is linear, and \(T\) is non-linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=-3 \, g\left(x\right) g'\left(x\right) - 2 \, g\left(x\right)\hspace{3em}T(g(x))=-3 \, x^{3} g\left(x\right) - g'\left(x\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=-x^{3} - 2 \, f'\left(x\right)\hspace{3em}T(f(x))=5 \, x^{3} f\left(x\right) + 3 \, f\left(x^{2}\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=-h\left(x\right)^{3} + h\left(-2\right)\hspace{3em}T(h(x))=4 \, h\left(x^{2}\right) - 3 \, h'\left(x\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=2 \, x^{2} + 4 \, h'\left(1\right)\hspace{3em}T(h(x))=5 \, x^{2} h\left(x\right) + 4 \, h\left(-2\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=5 \, f\left(1\right) - f'\left(x\right)\hspace{3em}T(f(x))=-3 \, f\left(x\right)^{3} + 2 \, f\left(x\right)\]

\(S\) is linear, and \(T\) is non-linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=-2 \, h\left(-1\right) - 2 \, h'\left(x\right)\hspace{3em}T(h(x))=2 \, h\left(x\right) h'\left(x\right) - 5 \, h'\left(-5\right)\]

\(S\) is linear, and \(T\) is non-linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=4 \, g\left(x\right)^{3} + 3 \, g'\left(x\right)\hspace{3em}T(g(x))=2 \, x g\left(x\right) - 5 \, g'\left(1\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=x^{3} f\left(x\right) + 3 \, f'\left(x\right)\hspace{3em}T(f(x))=-4 \, x^{2} - 4 \, f\left(x^{3}\right)\]

\(S\) is linear, and \(T\) is non-linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=-2 \, g'\left(-1\right) - 3 \, g'\left(x\right)\hspace{3em}T(g(x))=2 \, x^{3} + 5 \, g\left(2\right)\]

\(S\) is linear, and \(T\) is non-linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=-f\left(x\right) f'\left(x\right) - 2 \, f'\left(x\right)\hspace{3em}T(f(x))=4 \, f\left(3\right) + 5 \, f\left(x^{2}\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=2 \, x + 5 \, f\left(x\right)\hspace{3em}T(f(x))=-3 \, f\left(-3\right) + 5 \, f'\left(x\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=-x^{2} h\left(x\right) + 4 \, h'\left(-2\right)\hspace{3em}T(h(x))=5 \, h\left(x\right) h'\left(x\right) + h\left(x\right)\]

\(S\) is linear, and \(T\) is non-linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=4 \, g\left(x\right)^{3} + 5 \, g\left(-5\right)\hspace{3em}T(g(x))=2 \, x^{3} g\left(x\right) + 2 \, g'\left(x\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=-h\left(x\right) h'\left(x\right) + 4 \, h'\left(x\right)\hspace{3em}T(h(x))=5 \, x h\left(x\right) + 5 \, h\left(-1\right)\]

\(S\) is non-linear, and \(T\) is linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=f\left(4\right) + f'\left(x\right)\hspace{3em}T(f(x))=5 \, f\left(x\right) f'\left(x\right) + 4 \, f'\left(1\right)\]

\(S\) is linear, and \(T\) is non-linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=5 \, g\left(-1\right) - 5 \, g\left(x\right)\hspace{3em}T(g(x))=x - 2 \, g'\left(4\right)\]

\(S\) is linear, and \(T\) is non-linear.

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=g\left(3\right) + 2 \, g\left(x^{3}\right)\hspace{3em}T(g(x))=-4 \, x^{2} - g\left(x\right)\]

\(S\) is linear, and \(T\) is non-linear.