Question

determine whether the following function is a polynomial function. if the function is a polynomial function, state its degree. if it is not, tell why not. write the polynomial in standard form. then identify the leading term and the constant term.

\\g(x) = 2(x - 1)^2(x^2 + 3)\\

part 1 of 2

determine whether \\(g(x)\\) is a polynomial or not. select the correct choice below and, if necessary, fill in the

Explanation:

Identify polynomial function type

The given function is:

$$ G(x) = 2(x - 1)^2(x^2 + 3) $$

A polynomial function is a function of the form \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \), where the coefficients are real numbers and the exponents are non-negative integers. Since \( G(x) \) is a product of real constants and terms with non-negative integer powers of \( x \), it is a polynomial function.

Expand the function expression

To write the polynomial in standard form, we expand the algebraic expression:

$$ (x - 1)^2 = x^2 - 2x + 1 $$

Now, multiply this result by \( (x^2 + 3) \):

$$ (x^2 - 2x + 1)(x^2 + 3) = x^2(x^2 + 3) - 2x(x^2 + 3) + 1(x^2 + 3) $$

$$ = x^4 + 3x^2 - 2x^3 - 6x + x^2 + 3 $$

Combine like terms:

$$ = x^4 - 2x^3 + 4x^2 - 6x + 3 $$

Multiply the entire expanded expression by the constant factor \( 2 \):

$$ G(x) = 2(x^4 - 2x^3 + 4x^2 - 6x + 3) $$

$$ G(x) = 2x^4 - 4x^3 + 8x^2 - 12x + 6 $$

Determine degree and terms

From the standard form \( G(x) = 2x^4 - 4x^3 + 8x^2 - 12x + 6 \):

• The highest power of \( x \) is \( 4 \), so the degree is \( 4 \).
• The term containing the highest power is the leading term, which is \( 2x^4 \).
• The constant term is the term without \( x \), which is \( 6 \).

Answer:

The function \( G(x) = 2(x - 1)^2(x^2 + 3) \) is a polynomial function.

• Standard Form: \( G(x) = 2x^4 - 4x^3 + 8x^2 - 12x + 6 \)
• Degree: \( 4 \)
• Leading Term: \( 2x^4 \)
• Constant Term: \( 6 \)