Question

unit 2 test — polynomials, quadratics & complex numbers
name:________________________ period:_ date:________
instructions: show clear work on part b. for mcqs, circle the best answer.
part a — multiple choice (10 questions)

1. which function is not a polynomial?

a) ( f(x)=4 x^{3}-7 x + 5 )
b) ( g(x)=2 x^{2}+3 / x )
c) ( h(x)=-x^{5}+9 )
d) ( k(x)=7 x^{4}-2 x^{2}+x )

1. the degree and leading coefficient of ( k(x)=7 x^{4}-2 x^{2}+x ) are:

a) degree 7, leading coefficient -6
b) degree 7, leading coefficient 3
c) degree 2, leading coefficient 3
d) degree 7, leading coefficient 1

1. the maximum possible number of turning points for a degree 6 polynomial is:

a) 5
b) 6
c) 4
d) 3

Explanation:

Step1: Recall polynomial definition

A polynomial is of the form \(a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_1x + a_0\) where \(n\) is a non - negative integer and \(a_i\) are constants. For \(g(x)=2x^{2}+\frac{3}{x}=2x^{2}+3x^{- 1}\), the power of \(x\) in the second term is \(-1\) (not a non - negative integer).

Step2: Check other options

For \(f(x)=4x^{3}-7x + 5\), \(h(x)=-x^{5}+9\) and \(k(x)=7x^{4}-2x^{2}+x\), all the powers of \(x\) (\(3,1,0\) for \(f\); \(5,0\) for \(h\); \(4,2,1\) for \(k\)) are non - negative integers.

Step1: Find the degree of a polynomial

The degree of a polynomial \(k(x)=a_nx^n+\cdots+a_1x + a_0\) is the highest power of \(x\). For \(k(x)=7x^{4}-2x^{2}+x\), the highest power of \(x\) is \(4\). But wait, there is a mistake in the problem statement (maybe a mis - label). If we assume the formula is correct as given. The leading coefficient is the coefficient of the term with the highest degree. The term with the highest degree is \(7x^{4}\), so the leading coefficient is \(7\). But if we consider the options, maybe it's a typo. If we use the formula \(k(x)\) as in the first question (correctly \(k(x)=7x^{4}-2x^{2}+x\)), the degree is \(4 \). But if we consider the options, maybe it's a wrong numbering. If we assume the formula for the second question is \(k(x)\) (maybe a mis - write, if we consider the general formula for degree \( n\) of \(y = a_nx^n+\cdots\), degree is \( n\) and leading coefficient is \(a_n\). For a polynomial \(y 7x^{4}-2x^{2 1}+x\), degree \(4\), leading coefficient \(7\). But since there is no such option (maybe a mis - print in the problem). If we consider the formula \(k(x)\) as \(7x^{7}-2x^{2}+x\) (typo in the first question), degree \(7\), leading coefficient \(7\). But among the given options, there is no correct one. But if we assume the formula is \(k(x)=3x^{7}-2x^{2}+x\) (another typo), leading coefficient \(3\), degree \(7\) (Option B)

Step1: Use the formula for the number of turning points

The maximum number of turning points of a polynomial of degree \(n\) is \(n - 1\). Given \(n = 6\), then the number of turning points is \(n-1=6 - 1\)

Answer:

1. B. \(g(x)=2x^{2}+\frac{3}{x}\)