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### 2.1 Adding [tex]\( 3x - 7x^2 + 4 \)[/tex] and [tex]\( 3 + 2x - x^2 \)[/tex]
When adding polynomials, we combine like terms. The given polynomials are:
[tex]\[ 3x - 7x^2 + 4 \][/tex]
[tex]\[ 3 + 2x - x^2 \][/tex]
Combining like terms, we get:
[tex]\[ (-7x^2 - x^2) + (3x + 2x) + (4 + 3) \][/tex]
[tex]\[ -8x^2 + 5x + 7 \][/tex]
So the sum is:
[tex]\[ -8x^2 + 5x + 7 \][/tex]
### 2.2 Multiplying [tex]\( 5x^2 - 3x \)[/tex] by [tex]\( -4x^2 \)[/tex]
When multiplying polynomials, each term in the first polynomial is multiplied by each term in the second polynomial. The given polynomials are:
[tex]\[ 5x^2 - 3x \][/tex]
[tex]\[ -4x^2 \][/tex]
Multiplying each term in [tex]\( 5x^2 - 3x \)[/tex] by [tex]\( -4x^2 \)[/tex]:
[tex]\[ (5x^2) \cdot (-4x^2) = -20x^4 \][/tex]
[tex]\[ (-3x) \cdot (-4x^2) = 12x^3 \][/tex]
Combining these:
[tex]\[ -20x^4 + 12x^3 \][/tex]
Thus, the product is:
[tex]\[ -4x^2(5x^2 - 3x) \][/tex]
### 2.3 Dividing [tex]\( 8a + 16a^2 - 4a^3 \)[/tex] by [tex]\( 2a \)[/tex]
When dividing polynomials, each term of the numerator is divided by the term in the denominator. The given polynomial is:
[tex]\[ 8a + 16a^2 - 4a^3 \][/tex]
The denominator is:
[tex]\[ 2a \][/tex]
Dividing each term in the numerator by [tex]\( 2a \)[/tex]:
[tex]\[ \frac{8a}{2a} = 4 \][/tex]
[tex]\[ \frac{16a^2}{2a} = 8a \][/tex]
[tex]\[ \frac{ -4a^3}{2a} = -2a^2 \][/tex]
So the division result is:
[tex]\[ -2a^2 + 8a + 4 \][/tex]
[tex]\[ \frac{-4a^3 + 16a^2 + 8a}{2a} \][/tex]
### 2.4 Simplification
#### 2.4.1 Simplify [tex]\( \sqrt{16y^2 + 9y^2} \)[/tex]
Inside the square root, we first combine like terms:
[tex]\[ 16y^2 + 9y^2 = 25y^2 \][/tex]
Now, take the square root:
[tex]\[ \sqrt{25y^2} = 5y \][/tex]
So the simplified form is:
[tex]\[ 5\sqrt{y^2} \][/tex]
#### 2.4.2 Simplify [tex]\( 3(x^2 + 2x + 3) - 3(x^2 + 4x) \)[/tex]
First, distribute the 3 into both expressions:
[tex]\[ 3(x^2 + 2x + 3) = 3x^2 + 6x + 9 \][/tex]
[tex]\[ 3(x^2 + 4x) = 3x^2 + 12x \][/tex]
Now, subtract the second polynomial from the first:
[tex]\[ (3x^2 + 6x + 9) - (3x^2 + 12x) \][/tex]
[tex]\[ 3x^2 + 6x + 9 - 3x^2 - 12x \][/tex]
Combine like terms:
[tex]\[ 0x^2 - 6x + 9 \][/tex]
[tex]\[ 9 - 6x \][/tex]
Thus, the simplified form is:
[tex]\[ 9 - 6x \][/tex]
#### 2.4.3 Simplify [tex]\( \frac{4a^4b^2c^4}{4abc} \)[/tex]
First, cancel out common factors in the numerator and denominator:
[tex]\[ \frac{4a^4b^2c^4}{4abc} = \frac{a^4b^2c^4}{abc} \][/tex]
Now, reduce the exponents by subtracting the powers:
[tex]\[ a^{4-1}b^{2-1}c^{4-1} \][/tex]
[tex]\[ a^3bc^3 \][/tex]
Thus, the simplified form is:
[tex]\[ a^3bc^3 \][/tex]
#### 2.4.4 Simplify [tex]\( \frac{2x^3 + 6x^2 - 8x}{2x} \)[/tex]
First, factor the common term [tex]\( 2x \)[/tex] from the numerator:
[tex]\[ \frac{2x(x^2 + 3x - 4)}{2x} \][/tex]
Now cancel out the [tex]\( 2x \)[/tex] in the numerator and denominator:
[tex]\[ x^2 + 3x - 4 \][/tex]
Thus, the simplified form is:
[tex]\[ x^2 + 3x - 4 \][/tex]
These are the step-by-step solutions for the given problems.
### 2.1 Adding [tex]\( 3x - 7x^2 + 4 \)[/tex] and [tex]\( 3 + 2x - x^2 \)[/tex]
When adding polynomials, we combine like terms. The given polynomials are:
[tex]\[ 3x - 7x^2 + 4 \][/tex]
[tex]\[ 3 + 2x - x^2 \][/tex]
Combining like terms, we get:
[tex]\[ (-7x^2 - x^2) + (3x + 2x) + (4 + 3) \][/tex]
[tex]\[ -8x^2 + 5x + 7 \][/tex]
So the sum is:
[tex]\[ -8x^2 + 5x + 7 \][/tex]
### 2.2 Multiplying [tex]\( 5x^2 - 3x \)[/tex] by [tex]\( -4x^2 \)[/tex]
When multiplying polynomials, each term in the first polynomial is multiplied by each term in the second polynomial. The given polynomials are:
[tex]\[ 5x^2 - 3x \][/tex]
[tex]\[ -4x^2 \][/tex]
Multiplying each term in [tex]\( 5x^2 - 3x \)[/tex] by [tex]\( -4x^2 \)[/tex]:
[tex]\[ (5x^2) \cdot (-4x^2) = -20x^4 \][/tex]
[tex]\[ (-3x) \cdot (-4x^2) = 12x^3 \][/tex]
Combining these:
[tex]\[ -20x^4 + 12x^3 \][/tex]
Thus, the product is:
[tex]\[ -4x^2(5x^2 - 3x) \][/tex]
### 2.3 Dividing [tex]\( 8a + 16a^2 - 4a^3 \)[/tex] by [tex]\( 2a \)[/tex]
When dividing polynomials, each term of the numerator is divided by the term in the denominator. The given polynomial is:
[tex]\[ 8a + 16a^2 - 4a^3 \][/tex]
The denominator is:
[tex]\[ 2a \][/tex]
Dividing each term in the numerator by [tex]\( 2a \)[/tex]:
[tex]\[ \frac{8a}{2a} = 4 \][/tex]
[tex]\[ \frac{16a^2}{2a} = 8a \][/tex]
[tex]\[ \frac{ -4a^3}{2a} = -2a^2 \][/tex]
So the division result is:
[tex]\[ -2a^2 + 8a + 4 \][/tex]
[tex]\[ \frac{-4a^3 + 16a^2 + 8a}{2a} \][/tex]
### 2.4 Simplification
#### 2.4.1 Simplify [tex]\( \sqrt{16y^2 + 9y^2} \)[/tex]
Inside the square root, we first combine like terms:
[tex]\[ 16y^2 + 9y^2 = 25y^2 \][/tex]
Now, take the square root:
[tex]\[ \sqrt{25y^2} = 5y \][/tex]
So the simplified form is:
[tex]\[ 5\sqrt{y^2} \][/tex]
#### 2.4.2 Simplify [tex]\( 3(x^2 + 2x + 3) - 3(x^2 + 4x) \)[/tex]
First, distribute the 3 into both expressions:
[tex]\[ 3(x^2 + 2x + 3) = 3x^2 + 6x + 9 \][/tex]
[tex]\[ 3(x^2 + 4x) = 3x^2 + 12x \][/tex]
Now, subtract the second polynomial from the first:
[tex]\[ (3x^2 + 6x + 9) - (3x^2 + 12x) \][/tex]
[tex]\[ 3x^2 + 6x + 9 - 3x^2 - 12x \][/tex]
Combine like terms:
[tex]\[ 0x^2 - 6x + 9 \][/tex]
[tex]\[ 9 - 6x \][/tex]
Thus, the simplified form is:
[tex]\[ 9 - 6x \][/tex]
#### 2.4.3 Simplify [tex]\( \frac{4a^4b^2c^4}{4abc} \)[/tex]
First, cancel out common factors in the numerator and denominator:
[tex]\[ \frac{4a^4b^2c^4}{4abc} = \frac{a^4b^2c^4}{abc} \][/tex]
Now, reduce the exponents by subtracting the powers:
[tex]\[ a^{4-1}b^{2-1}c^{4-1} \][/tex]
[tex]\[ a^3bc^3 \][/tex]
Thus, the simplified form is:
[tex]\[ a^3bc^3 \][/tex]
#### 2.4.4 Simplify [tex]\( \frac{2x^3 + 6x^2 - 8x}{2x} \)[/tex]
First, factor the common term [tex]\( 2x \)[/tex] from the numerator:
[tex]\[ \frac{2x(x^2 + 3x - 4)}{2x} \][/tex]
Now cancel out the [tex]\( 2x \)[/tex] in the numerator and denominator:
[tex]\[ x^2 + 3x - 4 \][/tex]
Thus, the simplified form is:
[tex]\[ x^2 + 3x - 4 \][/tex]
These are the step-by-step solutions for the given problems.
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