Question

simplify the given polynomials. then, classify each polynomial by its degree and number of terms.
polynomial 1 : \\(\left(x - \frac{1}{2}\
ight)\left(6x + 2\
ight)\\)
polynomial 2 : \\(\left(7x^2 + 3x\
ight) - \frac{1}{3}\left(21x^2 - 12\
ight)\\)
polynomial 3 : \\(4\left(5x^2 - 9x + 7\
ight) + 2\left(-10x^2 + 18x - 13\
ight)\\)

6 \quad binomial \quad constant \quad \\(6x^2 - x - 1\\) \quad linear \quad 2 \quad \\(6x^2 + 2x - \frac{1}{2}\\) \quad trinomial

polynomial	simplified form	name by degree	name by number of terms
2	\\(3x + 4\\)
3			monomial

Explanation:

Polynomial 1:

Step 1: Expand the product

Using the distributive property (FOIL method):

$$\begin{align*}(x - \frac{1}{2})(6x + 2) &= x(6x) + x(2) - \frac{1}{2}(6x) - \frac{1}{2}(2) \\ &= 6x^2 + 2x - 3x - 1 \\ &= 6x^2 - x - 1\end{align*}$$

Step 2: Classify by number of terms

The simplified form \(6x^2 - x - 1\) has 3 terms, so it is a trinomial.

Polynomial 2:

Step 1: Simplify the expression

First, distribute \(-\frac{1}{3}\) to \(21x^2 - 12\):

$$\begin{align*}(7x^2 + 3x) - \frac{1}{3}(21x^2 - 12) &= 7x^2 + 3x - 7x^2 + 4 \\ &= (7x^2 - 7x^2) + 3x + 4 \\ &= 3x + 4\end{align*}$$

Step 2: Classify by degree

The highest power of \(x\) is 1, so it is linear.

Step 3: Classify by number of terms

The simplified form \(3x + 4\) has 2 terms, so it is a binomial.

Polynomial 3:

Step 1: Expand and combine like terms

Distribute 4 and 2:

$$\begin{align*}4(5x^2 - 9x + 7) + 2(-10x^2 + 18x - 13) &= 20x^2 - 36x + 28 - 20x^2 + 36x - 26 \\ &= (20x^2 - 20x^2) + (-36x + 36x) + (28 - 26) \\ &= 0x^2 + 0x + 2 \\ &= 2\end{align*}$$

Step 2: Classify by degree

The highest power of \(x\) is 0 (since \(2 = 2x^0\)), so it is a constant polynomial.

Step 3: Classify by number of terms

The simplified form \(2\) has 1 term, so it is a monomial.

Final Table (Filled):

Polynomial	Simplified Form	Name by Degree	Name by Number of Terms
2	\(3x + 4\)	linear	binomial
3	\(2\)	constant	monomial

(If the question expects individual answers for each polynomial’s classification, here’s the breakdown:

• Polynomial 1: Simplified \(6x^2 - x - 1\), Degree: quadratic, Terms: trinomial
• Polynomial 2: Degree: linear, Terms: binomial
• Polynomial 3: Simplified \(2\), Degree: constant, Terms: monomial)

Answer:

Polynomial 1:

Step 1: Expand the product

Using the distributive property (FOIL method):

$$\begin{align*}(x - \frac{1}{2})(6x + 2) &= x(6x) + x(2) - \frac{1}{2}(6x) - \frac{1}{2}(2) \\ &= 6x^2 + 2x - 3x - 1 \\ &= 6x^2 - x - 1\end{align*}$$

Step 2: Classify by number of terms

The simplified form \(6x^2 - x - 1\) has 3 terms, so it is a trinomial.

Polynomial 2:

Step 1: Simplify the expression

First, distribute \(-\frac{1}{3}\) to \(21x^2 - 12\):

$$\begin{align*}(7x^2 + 3x) - \frac{1}{3}(21x^2 - 12) &= 7x^2 + 3x - 7x^2 + 4 \\ &= (7x^2 - 7x^2) + 3x + 4 \\ &= 3x + 4\end{align*}$$

Step 2: Classify by degree

The highest power of \(x\) is 1, so it is linear.

Step 3: Classify by number of terms

The simplified form \(3x + 4\) has 2 terms, so it is a binomial.

Polynomial 3:

Step 1: Expand and combine like terms

Distribute 4 and 2:

$$\begin{align*}4(5x^2 - 9x + 7) + 2(-10x^2 + 18x - 13) &= 20x^2 - 36x + 28 - 20x^2 + 36x - 26 \\ &= (20x^2 - 20x^2) + (-36x + 36x) + (28 - 26) \\ &= 0x^2 + 0x + 2 \\ &= 2\end{align*}$$

Step 2: Classify by degree

The highest power of \(x\) is 0 (since \(2 = 2x^0\)), so it is a constant polynomial.

Step 3: Classify by number of terms

The simplified form \(2\) has 1 term, so it is a monomial.

Final Table (Filled):

Polynomial	Simplified Form	Name by Degree	Name by Number of Terms
2	\(3x + 4\)	linear	binomial
3	\(2\)	constant	monomial

(If the question expects individual answers for each polynomial’s classification, here’s the breakdown:

• Polynomial 1: Simplified \(6x^2 - x - 1\), Degree: quadratic, Terms: trinomial
• Polynomial 2: Degree: linear, Terms: binomial
• Polynomial 3: Simplified \(2\), Degree: constant, Terms: monomial)