Polynomial Functions

[nycmath]

Determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. Write each polynomial function in standard form. Then identify the leading term and constant term.

See attachment.

[Math Tutor]

A. H(x) = √x(√x − 1) = x − √x = x − x^(1/2). Not a polynomial — the exponent 1/2 is not a nonnegative integer (and the domain is only x ≥ 0).

B. F(x) = (x² − 5)/x³ = x^(−1) − 5x^(−3). Not a polynomial — the variable sits in the denominator, i.e. the exponents are negative.

C. G(x) = −3x²(x + 2)³ is a polynomial. Since (x + 2)³ = x³ + 6x² + 12x + 8,

G(x) = −3x⁵ − 18x⁴ − 36x³ − 24x² (standard form)

Degree 5, leading term −3x⁵, constant term 0.

Quick check of the degree without expanding: 2 + 3 = 5.

[nycmath]

A. H(x) = √x(√x − 1) = x − √x = x − x^(1/2). Not a polynomial — the exponent 1/2 is not a nonnegative integer (and the domain is only x ≥ 0).

B. F(x) = (x² − 5)/x³ = x^(−1) − 5x^(−3). Not a polynomial — the variable sits in the denominator, i.e. the exponents are negative.

C. G(x) = −3x²(x + 2)³ is a polynomial. Since (x + 2)³ = x³ + 6x² + 12x + 8,

G(x) = −3x⁵ − 18x⁴ − 36x³ − 24x² (standard form)

Degree 5, leading term −3x⁵, constant term 0.

Quick check of the degree without expanding: 2 + 3 = 5.

Well-done. Good job.