For The Expression $5x^3 Y^3 + X Y^2 + 8$ To Be A Trinomial With A Degree Of 5, The Missing Exponent On The $x$-term Must Be _____.
Introduction
In algebra, a trinomial is a polynomial with three terms. The degree of a polynomial is the highest power of the variable in any of its terms. For the expression $5x^3 y^3 + x y^2 + 8$ to be a trinomial with a degree of 5, we need to find the missing exponent on the $x$-term. In this article, we will explore the concept of trinomials and polynomials, and provide a step-by-step solution to find the missing exponent.
Understanding Trinomials and Polynomials
A trinomial is a polynomial with three terms. The general form of a trinomial is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. The degree of a polynomial is the highest power of the variable in any of its terms. For example, in the polynomial $x^2 + 3x + 2$, the degree is 2, because the highest power of $x$ is 2.
The Expression Given
The expression given is $5x^3 y^3 + x y^2 + 8$. To be a trinomial with a degree of 5, the missing exponent on the $x$-term must be found.
Step 1: Identify the Degree of the Given Expression
The degree of the given expression is 3, because the highest power of $x$ is 3. To be a trinomial with a degree of 5, the missing exponent on the $x$-term must be 2, so that the degree of the expression becomes 5.
Step 2: Find the Missing Exponent
To find the missing exponent, we need to add the missing term to the expression. The missing term must have a degree of 2, so that the degree of the expression becomes 5. The missing term is $x^2$.
Conclusion
In conclusion, for the expression $5x^3 y^3 + x y^2 + 8$ to be a trinomial with a degree of 5, the missing exponent on the $x$-term must be 2. This is because the degree of the given expression is 3, and to be a trinomial with a degree of 5, the missing exponent on the $x$-term must be 2, so that the degree of the expression becomes 5.
Example
Let's consider an example to illustrate the concept. Suppose we have the expression $x^2 + 3x + 2$. To be a trinomial with a degree of 5, we need to find the missing exponent on the $x$-term. The degree of the given expression is 2, because the highest power of $x$ is 2. To be a trinomial with a degree of 5, the missing exponent on the $x$-term must be 3, so that the degree of the expression becomes 5.
Tips and Tricks
- To find the missing exponent, we need to add the missing term to the expression.
- The missing term must have a degree that is one less than the desired degree of the expression.
- The degree of a polynomial is the highest power of the variable in any of its terms.
Frequently Asked Questions
- What is a trinomial? A trinomial is a polynomial with three terms.
- What is the degree of a polynomial? The degree of a polynomial is the highest power of the variable in any of its terms.
- How do I find the missing exponent on the $x$-term? To find the missing exponent, we need to add the missing term to the expression. The missing term must have a degree that is one less than the desired degree of the expression.
References
Conclusion
Q: What is a trinomial?
A: A trinomial is a polynomial with three terms. The general form of a trinomial is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.
Q: What is the degree of a polynomial?
A: The degree of a polynomial is the highest power of the variable in any of its terms. For example, in the polynomial $x^2 + 3x + 2$, the degree is 2, because the highest power of $x$ is 2.
Q: How do I find the missing exponent on the x-term?
A: To find the missing exponent, we need to add the missing term to the expression. The missing term must have a degree that is one less than the desired degree of the expression.
Q: What is the desired degree of the expression?
A: The desired degree of the expression is 5.
Q: What is the degree of the given expression?
A: The degree of the given expression is 3.
Q: How do I find the missing exponent on the x-term to make the degree of the expression 5?
A: To find the missing exponent, we need to add the missing term to the expression. The missing term must have a degree that is one less than the desired degree of the expression. In this case, the missing term must have a degree of 2, so that the degree of the expression becomes 5.
Q: What is the missing term?
A: The missing term is $x^2$.
Q: Why is the missing term x^2?
A: The missing term is $x^2$ because it has a degree of 2, which is one less than the desired degree of the expression, which is 5.
Q: How do I add the missing term to the expression?
A: To add the missing term to the expression, we simply add $x^2$ to the expression.
Q: What is the resulting expression?
A: The resulting expression is $5x^3 y^3 + x y^2 + 8 + x^2$.
Q: What is the degree of the resulting expression?
A: The degree of the resulting expression is 5.
Q: Why is the degree of the resulting expression 5?
A: The degree of the resulting expression is 5 because the highest power of $x$ in the expression is 5.
Q: What is the missing exponent on the x-term?
A: The missing exponent on the x-term is 2.
Q: Why is the missing exponent on the x-term 2?
A: The missing exponent on the x-term is 2 because it is one less than the desired degree of the expression, which is 5.
Conclusion
In conclusion, to find the missing exponent on the x-term, we need to add the missing term to the expression. The missing term must have a degree that is one less than the desired degree of the expression. In this case, the missing term is $x^2$, and the missing exponent on the x-term is 2.