Question

circle the correct word/symbol to create true statements about polynomials.
if the leading term is positive | negative with an odd | even degree then, the end behavior is ↑ ↓
if the leading term is positive with an odd degree, then the end behavior is
identify the leading coefficient, degree, and the end behavior for the following polynomial functions below.

Explanation:

Step1: Determine the leading term, leading coefficient, and degree for each polynomial

• For the polynomial \(3x^{2}+6x - 10\):
• The leading term is the term with the highest degree. Here, the degree of \(3x^{2}\) is \(2\), the degree of \(6x\) is \(1\), and the degree of \(-10\) is \(0\). So the leading term is \(3x^{2}\), the leading coefficient is \(3\), and the degree is \(2\).
• For the end - behavior: Since the leading coefficient \(a = 3>0\) and the degree \(n = 2\) (even), as \(x

ightarrow\infty\), \(y = 3x^{2}+6x - 10
ightarrow\infty\) and as \(x
ightarrow-\infty\), \(y=3x^{2}+6x - 10
ightarrow\infty\) (because \(y = ax^{n}+\cdots\), and when \(n\) is even \(x^{n}\geq0\) for all \(x\), and \(a>0\)).

• For the polynomial \(-2x^{7}+6x^{5}+2x^{3}\):
• The leading term is \(-2x^{7}\) (since the degree of \(-2x^{7}\) is \(7\), the degree of \(6x^{5}\) is \(5\), and the degree of \(2x^{3}\) is \(3\)). The leading coefficient is \(-2\), and the degree is \(7\).
• For the end - behavior: Since the leading coefficient \(a=-2 < 0\) and the degree \(n = 7\) (odd), as \(x

ightarrow\infty\), \(y=-2x^{7}+6x^{5}+2x^{3}
ightarrow-\infty\) (because \(y = ax^{n}+\cdots\) and \(x^{n}>0\) when \(x>0\) and \(n\) odd, \(a<0\)) and as \(x
ightarrow-\infty\), \(x^{n}<0\) (when \(n\) odd), so \(y=-2x^{7}+6x^{5}+2x^{3}
ightarrow\infty\) (because \(y=ax^{n}+\cdots\) and \(a = - 2\), \(x^{n}<0\) for \(x<0\) and \(n\) odd, so \(ax^{n}>0\)).

• For the polynomial \(-4x^{4}-3x^{3}+x^{2}+4\):
• The leading term is \(-4x^{4}\) (degree of \(-4x^{4}\) is \(4\), degree of \(-3x^{3}\) is \(3\), degree of \(x^{2}\) is \(2\)). The leading coefficient is \(-4\), and the degree is \(4\).
• For the end - behavior: Since the leading coefficient \(a=-4<0\) and the degree \(n = 4\) (even), as \(x

ightarrow\infty\), \(y=-4x^{4}-3x^{3}+x^{2}+4
ightarrow-\infty\) (because \(y = ax^{n}+\cdots\) and \(x^{n}>0\) for \(x
eq0\) and \(n\) even, \(a<0\)) and as \(x
ightarrow-\infty\), \(y=-4x^{4}-3x^{3}+x^{2}+4
ightarrow-\infty\) (because \(x^{n}>0\) for \(n\) even and \(x<0\), \(a<0\)).

Answer:

Function	Leading Coefficient	Degree	End Behavior

ightarrow\pm\infty\), \(y
ightarrow\infty\) |

\(-2x^{7}+6x^{5}+2x^{3}\)

\(-2\)

\(7\)

As \(x

ightarrow\infty\), \(y
ightarrow-\infty\); as \(x
ightarrow-\infty\), \(y
ightarrow\infty\) |

\(-4x^{4}-3x^{3}+x^{2}+4\)

\(-4\)

\(4\)

As \(x

ightarrow\pm\infty\), \(y
ightarrow-\infty\) |