Answer :
Absolutely, let’s break down the expression [tex]\( 3x^2 - 2xy + y^3 \)[/tex] step-by-step to understand its components better.
### Step 1: Understand Each Term
The expression consists of three terms:
1. [tex]\( 3x^2 \)[/tex]
2. [tex]\( -2xy \)[/tex]
3. [tex]\( y^3 \)[/tex]
#### Term 1: [tex]\( 3x^2 \)[/tex]
- This term represents a quadratic term in [tex]\( x \)[/tex] multiplied by a constant 3.
- Here, [tex]\( x^2 \)[/tex] means [tex]\( x \)[/tex] is squared, and then the result is multiplied by 3.
#### Term 2: [tex]\( -2xy \)[/tex]
- This is a product of [tex]\( x \)[/tex] and [tex]\( y \)[/tex], multiplied by -2.
- It represents a linear interaction term between [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
#### Term 3: [tex]\( y^3 \)[/tex]
- This term represents a cubic term in [tex]\( y \)[/tex], which means [tex]\( y \)[/tex] is raised to the power of 3.
### Step 2: Combine All Terms
To form the full expression, we add all three terms together:
[tex]\[ 3x^2 - 2xy + y^3 \][/tex]
### Step 3: Evaluate at Specific Points (Example)
Let’s evaluate this expression for specific values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. Let’s take [tex]\( x = 1 \)[/tex] and [tex]\( y = 2 \)[/tex] as an example.
1. For [tex]\( x = 1 \)[/tex]:
- [tex]\( 3x^2 \)[/tex]
[tex]\[ 3(1)^2 = 3 \][/tex]
2. For [tex]\( y = 2 \)[/tex]:
- [tex]\(-2xy\)[/tex]
[tex]\[ -2(1)(2) = -4 \][/tex]
3. For [tex]\( y = 2 \)[/tex]:
- [tex]\( y^3 \)[/tex]
[tex]\[ (2)^3 = 8 \][/tex]
4. Combine these results:
[tex]\[ 3 - 4 + 8 = 7 \][/tex]
So, for [tex]\( x = 1 \)[/tex] and [tex]\( y = 2 \)[/tex], the expression evaluates to 7.
### Step 4: Significance of Each Term
This example helps us understand that the expression combines both the quadratic term in [tex]\( x \)[/tex], a mixed term involving both [tex]\( x \)[/tex] and [tex]\( y \)[/tex], and a cubic term in [tex]\( y \)[/tex]. The interplay of these terms will affect the overall value depending on the specific values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
### Conclusion
The expression [tex]\( 3x^2 - 2xy + y^3 \)[/tex] is a polynomial that can be analyzed by considering each term separately, understanding its contribution, and then combining them to form the complete polynomial expression.
### Step 1: Understand Each Term
The expression consists of three terms:
1. [tex]\( 3x^2 \)[/tex]
2. [tex]\( -2xy \)[/tex]
3. [tex]\( y^3 \)[/tex]
#### Term 1: [tex]\( 3x^2 \)[/tex]
- This term represents a quadratic term in [tex]\( x \)[/tex] multiplied by a constant 3.
- Here, [tex]\( x^2 \)[/tex] means [tex]\( x \)[/tex] is squared, and then the result is multiplied by 3.
#### Term 2: [tex]\( -2xy \)[/tex]
- This is a product of [tex]\( x \)[/tex] and [tex]\( y \)[/tex], multiplied by -2.
- It represents a linear interaction term between [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
#### Term 3: [tex]\( y^3 \)[/tex]
- This term represents a cubic term in [tex]\( y \)[/tex], which means [tex]\( y \)[/tex] is raised to the power of 3.
### Step 2: Combine All Terms
To form the full expression, we add all three terms together:
[tex]\[ 3x^2 - 2xy + y^3 \][/tex]
### Step 3: Evaluate at Specific Points (Example)
Let’s evaluate this expression for specific values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. Let’s take [tex]\( x = 1 \)[/tex] and [tex]\( y = 2 \)[/tex] as an example.
1. For [tex]\( x = 1 \)[/tex]:
- [tex]\( 3x^2 \)[/tex]
[tex]\[ 3(1)^2 = 3 \][/tex]
2. For [tex]\( y = 2 \)[/tex]:
- [tex]\(-2xy\)[/tex]
[tex]\[ -2(1)(2) = -4 \][/tex]
3. For [tex]\( y = 2 \)[/tex]:
- [tex]\( y^3 \)[/tex]
[tex]\[ (2)^3 = 8 \][/tex]
4. Combine these results:
[tex]\[ 3 - 4 + 8 = 7 \][/tex]
So, for [tex]\( x = 1 \)[/tex] and [tex]\( y = 2 \)[/tex], the expression evaluates to 7.
### Step 4: Significance of Each Term
This example helps us understand that the expression combines both the quadratic term in [tex]\( x \)[/tex], a mixed term involving both [tex]\( x \)[/tex] and [tex]\( y \)[/tex], and a cubic term in [tex]\( y \)[/tex]. The interplay of these terms will affect the overall value depending on the specific values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
### Conclusion
The expression [tex]\( 3x^2 - 2xy + y^3 \)[/tex] is a polynomial that can be analyzed by considering each term separately, understanding its contribution, and then combining them to form the complete polynomial expression.
