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]
