Understanding Trinomials and Polynomials

Oct 13, 2025 by windailys.com 35 views

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.