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Drag each number to the correct location on the table. Each number can be used more than once, but not all numbers will be used.

Simplify the given polynomial expressions, and determine the degree and number of terms in each expression.

Numbers: 4, 3, 31, 6, 0, 2, 5

Given Polynomial Expressions:
[tex]\[
\begin{array}{c}
4x + 2x^2(3x - 5) \\
\left(-3x^4 + 5x^3 - 12\right) + \left(7x^3 - x^5 + 6\right) \\
\left(3x^2 - 3\right)\left(3x^2 + 3\right)
\end{array}
\][/tex]

Number of Terms:
[tex]\[
\begin{array}{c}
4x + 2x^2(3x - 5) \\
\left(-3x^4 + 5x^3 - 12\right) + \left(7x^3 - x^5 + 6\right) \\
\left(3x^2 - 3\right)\left(3x^2 + 3\right)
\end{array}
\][/tex]

Degree:
[tex]\[
\begin{array}{c}
4x + 2x^2(3x - 5) \\
\left(-3x^4 + 5x^3 - 12\right) + \left(7x^3 - x^5 + 6\right) \\
\left(3x^2 - 3\right)\left(3x^2 + 3\right)
\end{array}
\][/tex]



Answer :

GinnyAnswer
Let's go through each polynomial expression step-by-step, simplify them, and determine their degrees and the number of terms.

Given expressions:
1. [tex]\(4x + 2x^2(3x - 5)\)[/tex]
2. [tex]\((-3x^4 + 5x^3 - 12) + (7x^3 - x^5 + 6)\)[/tex]
3. [tex]\((3x^2 - 3)(3x^2 + 3)\)[/tex]

### Simplifying the Expressions:

1. Expression 1: [tex]\(4x + 2x^2(3x - 5)\)[/tex]
- Start by expanding the term inside the parentheses: [tex]\(2x^2(3x - 5) = 6x^3 - 10x^2\)[/tex]
- Now combine it with [tex]\(4x\)[/tex]:
[tex]\[ 4x + 6x^3 - 10x^2 \][/tex]
- The simplified form is [tex]\(6x^3 - 10x^2 + 4x\)[/tex].
- Degree of the polynomial: The highest power is [tex]\(3\)[/tex].
- Number of terms: There are [tex]\(3\)[/tex] terms.

2. Expression 2: [tex]\((-3x^4 + 5x^3 - 12) + (7x^3 - x^5 + 6)\)[/tex]
- Combine like terms:
[tex]\[ (-3x^4 + 5x^3 - 12) + (7x^3 - x^5 + 6) = -x^5 - 3x^4 + 5x^3 + 7x^3 - 12 + 6 \][/tex]
[tex]\[ = -x^5 - 3x^4 + 12x^3 - 6 \][/tex]
- The simplified form is [tex]\(-x^5 - 3x^4 + 12x^3 - 6\)[/tex].
- Degree of the polynomial: The highest power is [tex]\(5\)[/tex].
- Number of terms: There are [tex]\(4\)[/tex] terms.

3. Expression 3: [tex]\((3x^2 - 3)(3x^2 + 3)\)[/tex]
- Expand the product using the distributive property or FOIL method:
[tex]\[ (3x^2 - 3)(3x^2 + 3) = 3x^2 \cdot 3x^2 + 3x^2 \cdot 3 - 3 \cdot 3x^2 - 3 \cdot 3 \][/tex]
[tex]\[ = 9x^4 + 9x^2 - 9x^2 - 9 \][/tex]
[tex]\[ = 9x^4 - 9 \][/tex]
- The simplified form is [tex]\(9x^4 - 9\)[/tex].
- Degree of the polynomial: The highest power is [tex]\(4\)[/tex].
- Number of terms: There are [tex]\(2\)[/tex] terms.

### Summary of Results
Let's fill in the table with the degree and the number of terms for each polynomial.

[tex]\[ \begin{array}{ccc} \text{Expression} & \text{Degree} & \text{Number of Terms} \\ 4x + 2x^2(3x - 5) & 3 & 3 \\ \left(-3x^4 + 5x^3 - 12\right) + \left(7x^3 - x^5 + 6\right) & 5 & 4 \\ (3x^2 - 3)(3x^2 + 3) & 4 & 2 \\ \end{array} \][/tex]

### Placing Numbers:

Number of Terms

[tex]\[ \begin{array}{c} 2 \\ 4 \\ 2 \end{array} \][/tex]

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