Let's simplify each polynomial step-by-step and classify them according to their degree and number of terms.
### Polynomial 1: [tex]\((3x-\frac{1}{4})(4x+8)\)[/tex]
To simplify Polynomial 1, we'll use the distributive property (also known as FOIL for binomials):
1. First term: [tex]\( 3x \cdot 4x = 12x^2 \)[/tex]
2. Outer term: [tex]\( 3x \cdot 8 = 24x \)[/tex]
3. Inner term: [tex]\( -\frac{1}{4} \cdot 4x = -x \)[/tex]
4. Last term: [tex]\( -\frac{1}{4} \cdot 8 = -2 \)[/tex]
Combine these terms:
[tex]\[ 12x^2 + 24x - x - 2 = 12x^2 + 23x - 2 \][/tex]
This is our simplified Polynomial 1. It is a quadratic (degree 2) polynomial with three terms, also known as a trinomial.
### Polynomial 2: [tex]\(\left(5 x^2+7 x\right)-\frac{1}{2}\left(10 x^2-4\right)\)[/tex]
To simplify Polynomial 2, we first distribute and then combine like terms:
1. Distribute [tex]\(\frac{1}{2}\)[/tex] into [tex]\((10 x^2 - 4)\)[/tex]:
[tex]\[ \frac{1}{2}(10 x^2 - 4) = 5 x^2 - 2 \][/tex]
2. Subtract this result from the original polynomial:
[tex]\[ (5 x^2 + 7 x) - (5 x^2 - 2) \][/tex]
Combine the like terms:
[tex]\[ 5 x^2 + 7 x - 5 x^2 + 2 = 7 x + 2 \][/tex]
This is our simplified Polynomial 2. It is a linear (degree 1) polynomial with two terms, also known as a binomial.
### Polynomial 3: [tex]\(3(8 x^2 + 4 x - 2) + 6(-4 x^2 - 2 x + 3)\)[/tex]
To simplify Polynomial 3, distribute the constants and then combine like terms:
1. Distribute the 3 into [tex]\((8 x^2 + 4 x - 2)\)[/tex]:
[tex]\[ 3(8 x^2 + 4 x - 2) = 24 x^2 + 12 x - 6 \][/tex]
2. Distribute the 6 into [tex]\((-4 x^2 - 2 x + 3)\)[/tex]:
[tex]\[ 6(-4 x^2 - 2 x + 3) = -24 x^2 - 12 x + 18 \][/tex]
3. Combine the like terms:
[tex]\[ 24 x^2 + 12 x - 6 + (-24 x^2 - 12 x + 18) = 24x^2 - 24x^2 + 12x - 12x -6 + 18 \ = 0x^2 + 0x + 12 = 12 \][/tex]
This is our simplified Polynomial 3. It is a constant (degree 0) polynomial with one term, also known as a monomial.
### Final Classification
Let's fill in the table with the simplified forms and classifications:
\begin{tabular}{|c|c|c|c|}
\hline Polynomial & Simplified Form & \begin{tabular}{c}
Name by \\
Degree
\end{tabular} & \begin{tabular}{c}
Name by \\
Number of Terms
\end{tabular} \\
\hline Polynomial 1 & [tex]\( 12 x^2 + 23 x - 2 \)[/tex] & quadratic & trinomial \\
\hline Polynomial 2 & [tex]\( 7x + 2 \)[/tex] & linear & binomial \\
\hline Polynomial 3 & 12 & constant & monomial \\
\hline
\end{tabular}
