Question

3.3.a independent practice & hw

1. classify each polynomial below by degree and number of terms. then, determine the degree, leading coefficient, and end behavior of each polynomial function below.

a. $f(x) = 8x^2 - 2x + 9x^3 - 2$
classification:
degree: leading coefficient:
end behavior:

b. $g(x) = 9x^3 - 10x^4 - 4x^2 + 7x - 1$
classification:
degree: leading coefficient:
end behavior:

c. $h(x) = 17x^2 - 1$
classification:
degree: leading coefficient:
end behavior:

d. $k(x) = 7x - 3 + 4x^2 - 12x^9$
classification:
degree: leading coefficient:
end behavior:

e. $y = (2x - 5)(3x + 4)$
classification:
degree: leading coefficient:
end behavior:

f. $y = -12(x - 1)(x - 3)^2$
classification:
degree: leading coefficient:
end behavior:

Explanation:

Part a: \( f(x) = 8x^2 - 2x + 9x^3 - 2 \)

Step 1: Simplify the polynomial

First, we arrange the terms in descending order of exponents: \( f(x) = 9x^3 + 8x^2 - 2x - 2 \)

Step 2: Classify by degree and number of terms

• Degree: The highest exponent is 3, so it's a cubic polynomial.
• Number of terms: There are 4 terms, so it's a quartic? Wait, no. Wait, cubic is degree 3, and the number of terms: 4 terms, so it's a cubic polynomial with 4 terms (a cubic polynomial, or a 4 - term polynomial, but classification by degree and number of terms: cubic (degree 3) and 4 terms (a quartic? No, cubic is degree 3. So classification: cubic polynomial (degree 3) with 4 terms (or a 4 - term cubic polynomial).

Step 3: Find the degree

The highest power of \( x \) is 3, so degree is 3.

Step 4: Find the leading coefficient

The leading term is \( 9x^3 \), so the leading coefficient is 9.

Step 5: Determine end behavior

For a polynomial \( a_nx^n+\cdots+a_0 \), as \( x
ightarrow\infty \) and \( x
ightarrow-\infty \):

• If \( n \) is odd and \( a_n>0 \), then as \( x

ightarrow\infty \), \( f(x)
ightarrow\infty \); as \( x
ightarrow-\infty \), \( f(x)
ightarrow-\infty \)
Here, \( n = 3 \) (odd) and \( a_n=9>0 \), so end behavior: as \( x
ightarrow\infty \), \( f(x)
ightarrow\infty \); as \( x
ightarrow-\infty \), \( f(x)
ightarrow-\infty \)

Part b: \( g(x)=9x^3 - 10x^4 - 4x^2+7x - 1 \)

Step 1: Simplify the polynomial

Arrange in descending order: \( g(x)=- 10x^4 + 9x^3-4x^2 + 7x - 1 \)

Step 2: Classify by degree and number of terms

• Degree: Highest exponent is 4, so it's a quartic polynomial.
• Number of terms: 5 terms, so a 5 - term quartic polynomial.

Step 3: Find the degree

Highest power of \( x \) is 4, so degree is 4.

Step 4: Find the leading coefficient

Leading term is \( - 10x^4 \), so leading coefficient is - 10.

Step 5: Determine end behavior

For a polynomial \( a_nx^n+\cdots+a_0 \), \( n = 4 \) (even) and \( a_n=-10<0 \)

• As \( x

ightarrow\infty \), \( f(x)
ightarrow-\infty \) (because \( (-10)(\infty)^4=-\infty \))

• As \( x

ightarrow-\infty \), \( f(x)
ightarrow-\infty \) (because \( (-10)(-\infty)^4=(-10)(\infty)=-\infty \))

Part c: \( h(x)=17x^2 - 1 \)

Step 1: Classify by degree and number of terms

• Degree: Highest exponent is 2, so it's a quadratic polynomial.
• Number of terms: 2 terms, so a binomial (2 - term) quadratic polynomial.

Step 2: Find the degree

Highest power of \( x \) is 2, so degree is 2.

Step 3: Find the leading coefficient

Leading term is \( 17x^2 \), so leading coefficient is 17.

Step 4: Determine end behavior

For \( n = 2 \) (even) and \( a_n = 17>0 \)

• As \( x

ightarrow\infty \), \( h(x)
ightarrow\infty \)

• As \( x

ightarrow-\infty \), \( h(x)
ightarrow\infty \) (because \( x^2 \) is positive for large \( |x| \))

Part d: \( k(x)=7x - 3 + 4x^2-12x^5 \)

Answer:

Step 1: Expand the polynomial

First, expand \( (x - 3)^2=x^2-6x + 9 \)
Then, \( (x - 1)(x^2-6x + 9)=x(x^2-6x + 9)-1(x^2-6x + 9)=x^3-6x^2+9x-x^2 + 6x-9=x^3-7x^2+15x - 9 \)
Then, \( y=-12(x^3-7x^2+15x - 9)=-12x^3 + 84x^2-180x + 108 \)

Step 2: Classify by degree and number of terms

• Degree: Highest exponent is 3, so it's a cubic polynomial.
• Number of terms: 4 terms, so a 4 - term cubic polynomial.

Step 3: Find the degree

Highest power of \( x \) is 3, so degree is 3.

Step 4: Find the leading coefficient

Leading term is \( - 12x^3 \), so leading coefficient is - 12.

Step 5: Determine end behavior

For \( n = 3 \) (odd) and \( a_n=-12<0 \)

• As \( x

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

• As \( x

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

Final Answers (Summarized)

Part a:

• Classification: Cubic polynomial (degree 3) with 4 terms
• Degree: 3
• Leading coefficient: 9
• End Behavior: As \( x

ightarrow\infty \), \( f(x)
ightarrow\infty \); as \( x
ightarrow-\infty \), \( f(x)
ightarrow-\infty \)

Part b:

• Classification: Quartic polynomial (degree 4) with 5 terms
• Degree: 4
• Leading coefficient: - 10
• End Behavior: As \( x

ightarrow\pm\infty \), \( g(x)
ightarrow-\infty \)

Part c:

• Classification: Quadratic polynomial (degree 2) with 2 terms (binomial)
• Degree: 2
• Leading coefficient: 17
• End Behavior: As \( x

ightarrow\pm\infty \), \( h(x)
ightarrow\infty \)

Part d:

• Classification: Quintic polynomial (degree 5) with 4 terms
• Degree: 5
• Leading coefficient: - 12
• End Behavior: As \( x

ightarrow\infty \), \( k(x)
ightarrow-\infty \); as \( x
ightarrow-\infty \), \( k(x)
ightarrow\infty \)

Part e:

• Classification: Quadratic polynomial (degree 2) with 3 terms (trinomial)
• Degree: 2
• Leading coefficient: 6
• End Behavior: As \( x

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

Part f:

• Classification: Cubic polynomial (degree 3) with 4 terms
• Degree: 3
• Leading coefficient: - 12
• End Behavior: As \( x

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