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Sagot :
Let's simplify each given polynomial step by step and classify them according to their degree and number of terms.
### Polynomial 1: [tex]\(\left(x - \frac{1}{2}\right)(6x + 2)\)[/tex]
1. Expand the expression:
[tex]\[ \left(x - \frac{1}{2}\right)(6x + 2) = x(6x + 2) - \frac{1}{2}(6x + 2) \][/tex]
2. Distribute:
[tex]\[ x(6x + 2) - \frac{1}{2}(6x + 2) = 6x^2 + 2x - 3x - 1 \][/tex]
3. Combine like terms:
[tex]\[ 6x^2 + 2x - 3x - 1 = 6x^2 - x - 1 \][/tex]
So, the simplified form of Polynomial 1 is [tex]\(6x^2 - x - 1\)[/tex].
- Degree: The highest power of [tex]\(x\)[/tex] is 2, so it is a quadratic polynomial.
- Number of Terms: There are 3 terms ( [tex]\(6x^2\)[/tex], [tex]\(-x\)[/tex], [tex]\(-1\)[/tex] ), so it is a trinomial.
### Polynomial 2: [tex]\(\left(7x^2 + 3x\right) - \frac{1}{3}\left(21x^2 - 12\right)\)[/tex]
1. Simplify inside the parentheses:
[tex]\[ \frac{1}{3}(21x^2 - 12) = 7x^2 - 4 \][/tex]
2. Combine like terms:
[tex]\[ \left(7x^2 + 3x\right) - (7x^2 - 4) = 7x^2 + 3x - 7x^2 + 4 \][/tex]
3. Combine like terms:
[tex]\[ 3x + 4 \][/tex]
So, the simplified form of Polynomial 2 is [tex]\(3x + 4\)[/tex].
- Degree: The highest power of [tex]\(x\)[/tex] is 1, so it is a linear polynomial.
- Number of Terms: There are 2 terms ( [tex]\(3x\)[/tex], [tex]\(4\)[/tex] ), so it is a binomial.
### Polynomial 3: [tex]\(4\left(5x^2 - 9x + 7\right) + 2\left(-10x^2 + 18x - 13\right)\)[/tex]
1. Distribute:
[tex]\[ 4(5x^2 - 9x + 7) + 2(-10x^2 + 18x - 13) = 20x^2 - 36x + 28 - 20x^2 + 36x - 26 \][/tex]
2. Combine like terms:
[tex]\[ 20x^2 - 20x^2 - 36x + 36x + 28 - 26 = 0x^2 + 0x + 2 = 2 \][/tex]
So, the simplified form of Polynomial 3 is [tex]\(2\)[/tex].
- Degree: There is no variable term, so it is a constant polynomial.
- Number of Terms: There is 1 term ( [tex]\(2\)[/tex] ), so it is a monomial.
### Summary Table
| Polynomial | Simplified Form | Name by Degree | Name by Number of Terms |
|------------|------------------|-------------------|-------------------------|
| 1 | [tex]\(6x^2 - x - 1\)[/tex] | quadratic | trinomial |
| 2 | [tex]\(3x + 4\)[/tex] | linear | binomial |
| 3 | [tex]\(2\)[/tex] | constant | monomial |
### Polynomial 1: [tex]\(\left(x - \frac{1}{2}\right)(6x + 2)\)[/tex]
1. Expand the expression:
[tex]\[ \left(x - \frac{1}{2}\right)(6x + 2) = x(6x + 2) - \frac{1}{2}(6x + 2) \][/tex]
2. Distribute:
[tex]\[ x(6x + 2) - \frac{1}{2}(6x + 2) = 6x^2 + 2x - 3x - 1 \][/tex]
3. Combine like terms:
[tex]\[ 6x^2 + 2x - 3x - 1 = 6x^2 - x - 1 \][/tex]
So, the simplified form of Polynomial 1 is [tex]\(6x^2 - x - 1\)[/tex].
- Degree: The highest power of [tex]\(x\)[/tex] is 2, so it is a quadratic polynomial.
- Number of Terms: There are 3 terms ( [tex]\(6x^2\)[/tex], [tex]\(-x\)[/tex], [tex]\(-1\)[/tex] ), so it is a trinomial.
### Polynomial 2: [tex]\(\left(7x^2 + 3x\right) - \frac{1}{3}\left(21x^2 - 12\right)\)[/tex]
1. Simplify inside the parentheses:
[tex]\[ \frac{1}{3}(21x^2 - 12) = 7x^2 - 4 \][/tex]
2. Combine like terms:
[tex]\[ \left(7x^2 + 3x\right) - (7x^2 - 4) = 7x^2 + 3x - 7x^2 + 4 \][/tex]
3. Combine like terms:
[tex]\[ 3x + 4 \][/tex]
So, the simplified form of Polynomial 2 is [tex]\(3x + 4\)[/tex].
- Degree: The highest power of [tex]\(x\)[/tex] is 1, so it is a linear polynomial.
- Number of Terms: There are 2 terms ( [tex]\(3x\)[/tex], [tex]\(4\)[/tex] ), so it is a binomial.
### Polynomial 3: [tex]\(4\left(5x^2 - 9x + 7\right) + 2\left(-10x^2 + 18x - 13\right)\)[/tex]
1. Distribute:
[tex]\[ 4(5x^2 - 9x + 7) + 2(-10x^2 + 18x - 13) = 20x^2 - 36x + 28 - 20x^2 + 36x - 26 \][/tex]
2. Combine like terms:
[tex]\[ 20x^2 - 20x^2 - 36x + 36x + 28 - 26 = 0x^2 + 0x + 2 = 2 \][/tex]
So, the simplified form of Polynomial 3 is [tex]\(2\)[/tex].
- Degree: There is no variable term, so it is a constant polynomial.
- Number of Terms: There is 1 term ( [tex]\(2\)[/tex] ), so it is a monomial.
### Summary Table
| Polynomial | Simplified Form | Name by Degree | Name by Number of Terms |
|------------|------------------|-------------------|-------------------------|
| 1 | [tex]\(6x^2 - x - 1\)[/tex] | quadratic | trinomial |
| 2 | [tex]\(3x + 4\)[/tex] | linear | binomial |
| 3 | [tex]\(2\)[/tex] | constant | monomial |
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