Question

answer the following questions using what youve learned from this unit. write your responses in the space provided.

1. part i: the degree of a polynomial is the (greatest / least) of the degrees of its terms. (circle the term that correctly completes this definition.) (1 point)

part ii: in order to write a polynomial in descending order, you must write the terms with the exponents (decreasing / increasing) from left to right. (circle the term that correctly completes this rule.) (1 point)
part iii: for each polynomial, determine the degree and write the polynomial in descending order. (4 points: 2 points each)
a. $-4x^{2}-12 + 11x^{4}$ b. $2x^{3}+14 - 3x^{4}+7x + 3x^{2}$

1. use addition and subtraction to simplify the following polynomials.

a. add polynomials $(3 - 4x + 8x^{2})+(-6 + 2x - 5x^{2})$
step 1: rewrite the polynomials without the parentheses. (1 point)
step 2: write the polynomial in descending order and use parentheses around like terms. (1 point)
step 3: add the like terms identified in step 2 to simplify the polynomial. (1 point)

Explanation:

Part I:

The degree of a polynomial is determined by the term with the highest - degree. So, the degree of a polynomial is the greatest of the degrees of its terms.

Part II:

When writing a polynomial in descending order, we arrange the terms from the term with the highest exponent to the term with the lowest exponent. So, we write the terms with the exponents decreasing from left - to - right.

Part III:

For polynomial \(A=-4x^{2}-12 + 11x^{4}\)

• Degree:

The degree of each term: For \(-4x^{2}\), the degree is \(2\); for \(-12\) (which can be written as \(-12x^{0}\)), the degree is \(0\); for \(11x^{4}\), the degree is \(4\). The degree of the polynomial is \(4\).

• Descending order:

We arrange the terms based on the exponents. So, \(11x^{4}-4x^{2}-12\)

For polynomial \(B = 2x^{3}+14-3x^{4}+7x + 3x^{2}\)

• Degree:

The degree of each term: For \(2x^{3}\), the degree is \(3\); for \(14\) (or \(14x^{0}\)), the degree is \(0\); for \(-3x^{4}\), the degree is \(4\); for \(7x\) (or \(7x^{1}\)), the degree is \(1\); for \(3x^{2}\), the degree is \(2\). The degree of the polynomial is \(4\).

• Descending order:

We arrange the terms based on the exponents. So, \(-3x^{4}+2x^{3}+3x^{2}+7x + 14\)

2.

Step 1:

Using the rule \(a+(b + c)=a + b + c\), \((3-4x + 8x^{2})+(-6 + 2x-5x^{2})=3-4x + 8x^{2}-6 + 2x-5x^{2}\)

Step 2:

Write in descending order: \((8x^{2}-5x^{2})+(-4x + 2x)+(3 - 6)\)

Step 3:

Combine like terms:
For the \(x^{2}\) terms: \(8x^{2}-5x^{2}=(8 - 5)x^{2}=3x^{2}\)
For the \(x\) terms: \(-4x+2x=(-4 + 2)x=-2x\)
For the constant terms: \(3-6=-3\)
The simplified polynomial is \(3x^{2}-2x - 3\)

Answer:

1.

• Part I: greatest
• Part II: decreasing
• Part III:
• A. Degree: \(4\), Descending order: \(11x^{4}-4x^{2}-12\)
• B. Degree: \(4\), Descending order: \(-3x^{4}+2x^{3}+3x^{2}+7x + 14\)

2.

• Step 1: \(3-4x + 8x^{2}-6 + 2x-5x^{2}\)
• Step 2: \((8x^{2}-5x^{2})+(-4x + 2x)+(3 - 6)\)
• Step 3: \(3x^{2}-2x - 3\)