Fill In The Blanks So That The Resulting Statement Is True.The Degree Of The Polynomial Function $f(x)=-2x^3(x-1)(x+5)$ Is $\square$. The Leading Coefficient Is $\square$ $\square$.

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Understanding Polynomial Functions

Polynomial functions are a fundamental concept in algebra, and they play a crucial role in various mathematical and real-world applications. A polynomial function is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication, and non-negative integer exponents. In this article, we will focus on determining the degree and leading coefficient of a given polynomial function.

Degree of a Polynomial Function

The degree of a polynomial function is the highest power or exponent of the variable in the polynomial. It is a measure of the complexity or the number of terms in the polynomial. For example, in the polynomial function f(x)=2x3+3x2−4x+1f(x) = 2x^3 + 3x^2 - 4x + 1, the degree is 3 because the highest power of the variable xx is 3.

Leading Coefficient

The leading coefficient of a polynomial function is the coefficient of the term with the highest power or degree. In the polynomial function f(x)=2x3+3x2−4x+1f(x) = 2x^3 + 3x^2 - 4x + 1, the leading coefficient is 2 because it is the coefficient of the term with the highest power, which is 2x32x^3.

Determining the Degree and Leading Coefficient of the Given Polynomial Function

The given polynomial function is $f(x)=-2x^3(x-1)(x+5)$. To determine the degree and leading coefficient of this function, we need to expand the expression and identify the term with the highest power.

Expanding the Polynomial Function

To expand the polynomial function, we need to multiply the terms together. We can start by multiplying the first two terms, −2x3-2x^3 and (x−1)(x-1).

f(x)=−2x3(x−1)(x+5)=−2x3(x2+4x−5)=−2x3x2−8x4+10x3\begin{aligned} f(x) &= -2x^3(x-1)(x+5) \\ &= -2x^3(x^2 + 4x - 5) \\ &= -2x^3x^2 - 8x^4 + 10x^3 \end{aligned}

Simplifying the Expression

Now that we have expanded the polynomial function, we can simplify the expression by combining like terms.

f(x)=−2x5−8x4+10x3=−2x5+(−8x4)+10x3\begin{aligned} f(x) &= -2x^5 - 8x^4 + 10x^3 \\ &= -2x^5 + (-8x^4) + 10x^3 \end{aligned}

Determining the Degree and Leading Coefficient

From the simplified expression, we can see that the term with the highest power is −2x5-2x^5. Therefore, the degree of the polynomial function is 5. The leading coefficient is -2, which is the coefficient of the term with the highest power.

Conclusion

In conclusion, the degree of the polynomial function $f(x)=-2x^3(x-1)(x+5)$ is 5, and the leading coefficient is -2. We determined the degree and leading coefficient by expanding the polynomial function and identifying the term with the highest power.

Example Use Case

Determining the degree and leading coefficient of a polynomial function is an essential skill in mathematics and has various applications in real-world problems. For example, in physics, the degree of a polynomial function can represent the order of a differential equation, and the leading coefficient can represent the coefficient of the highest-order derivative.

Tips and Tricks

  • When determining the degree and leading coefficient of a polynomial function, it is essential to expand the expression and identify the term with the highest power.
  • Use the distributive property to multiply the terms together and simplify the expression.
  • Combine like terms to simplify the expression and identify the term with the highest power.

Common Mistakes

  • Failing to expand the polynomial function and identify the term with the highest power.
  • Not combining like terms to simplify the expression.
  • Misinterpreting the degree and leading coefficient of the polynomial function.

Real-World Applications

Determining the degree and leading coefficient of a polynomial function has various real-world applications, including:

  • Physics: Determining the order of a differential equation and the coefficient of the highest-order derivative.
  • Engineering: Designing and analyzing systems that involve polynomial functions.
  • Economics: Modeling economic systems and predicting future trends.

Conclusion

In conclusion, determining the degree and leading coefficient of a polynomial function is an essential skill in mathematics and has various real-world applications. By expanding the polynomial function and identifying the term with the highest power, we can determine the degree and leading coefficient of the function.

Q: What is the degree of a polynomial function?

A: The degree of a polynomial function is the highest power or exponent of the variable in the polynomial. It is a measure of the complexity or the number of terms in the polynomial.

Q: How do I determine the degree of a polynomial function?

A: To determine the degree of a polynomial function, you need to expand the expression and identify the term with the highest power. You can use the distributive property to multiply the terms together and simplify the expression.

Q: What is the leading coefficient of a polynomial function?

A: The leading coefficient of a polynomial function is the coefficient of the term with the highest power or degree. It is the coefficient of the term that determines the degree of the polynomial function.

Q: How do I determine the leading coefficient of a polynomial function?

A: To determine the leading coefficient of a polynomial function, you need to identify the term with the highest power and determine its coefficient. The leading coefficient is the coefficient of the term with the highest power.

Q: What is the difference between the degree and leading coefficient of a polynomial function?

A: The degree of a polynomial function is the highest power or exponent of the variable in the polynomial, while the leading coefficient is the coefficient of the term with the highest power. The degree determines the complexity of the polynomial function, while the leading coefficient determines the coefficient of the highest-order term.

Q: Can a polynomial function have a degree of zero?

A: Yes, a polynomial function can have a degree of zero. A polynomial function with a degree of zero is a constant function, which means that it has no variable terms.

Q: Can a polynomial function have a negative degree?

A: No, a polynomial function cannot have a negative degree. The degree of a polynomial function is always a non-negative integer.

Q: How do I determine the degree and leading coefficient of a polynomial function with multiple variables?

A: To determine the degree and leading coefficient of a polynomial function with multiple variables, you need to expand the expression and identify the term with the highest power. You can use the distributive property to multiply the terms together and simplify the expression.

Q: Can a polynomial function have a degree of one?

A: Yes, a polynomial function can have a degree of one. A polynomial function with a degree of one is a linear function, which means that it has only one variable term.

Q: Can a polynomial function have a degree of two?

A: Yes, a polynomial function can have a degree of two. A polynomial function with a degree of two is a quadratic function, which means that it has two variable terms.

Q: Can a polynomial function have a degree of three?

A: Yes, a polynomial function can have a degree of three. A polynomial function with a degree of three is a cubic function, which means that it has three variable terms.

Q: Can a polynomial function have a degree of four?

A: Yes, a polynomial function can have a degree of four. A polynomial function with a degree of four is a quartic function, which means that it has four variable terms.

Q: Can a polynomial function have a degree of five?

A: Yes, a polynomial function can have a degree of five. A polynomial function with a degree of five is a quintic function, which means that it has five variable terms.

Q: Can a polynomial function have a degree of six?

A: Yes, a polynomial function can have a degree of six. A polynomial function with a degree of six is a sextic function, which means that it has six variable terms.

Q: Can a polynomial function have a degree of seven?

A: Yes, a polynomial function can have a degree of seven. A polynomial function with a degree of seven is a septic function, which means that it has seven variable terms.

Q: Can a polynomial function have a degree of eight?

A: Yes, a polynomial function can have a degree of eight. A polynomial function with a degree of eight is an octic function, which means that it has eight variable terms.

Q: Can a polynomial function have a degree of nine?

A: Yes, a polynomial function can have a degree of nine. A polynomial function with a degree of nine is a nonic function, which means that it has nine variable terms.

Q: Can a polynomial function have a degree of ten?

A: Yes, a polynomial function can have a degree of ten. A polynomial function with a degree of ten is a decic function, which means that it has ten variable terms.

Q: Can a polynomial function have a degree of eleven?

A: Yes, a polynomial function can have a degree of eleven. A polynomial function with a degree of eleven is an undecic function, which means that it has eleven variable terms.

Q: Can a polynomial function have a degree of twelve?

A: Yes, a polynomial function can have a degree of twelve. A polynomial function with a degree of twelve is a dodecic function, which means that it has twelve variable terms.

Q: Can a polynomial function have a degree of thirteen?

A: Yes, a polynomial function can have a degree of thirteen. A polynomial function with a degree of thirteen is a tridecic function, which means that it has thirteen variable terms.

Q: Can a polynomial function have a degree of fourteen?

A: Yes, a polynomial function can have a degree of fourteen. A polynomial function with a degree of fourteen is a tetradecic function, which means that it has fourteen variable terms.

Q: Can a polynomial function have a degree of fifteen?

A: Yes, a polynomial function can have a degree of fifteen. A polynomial function with a degree of fifteen is a pentadecic function, which means that it has fifteen variable terms.

Q: Can a polynomial function have a degree of sixteen?

A: Yes, a polynomial function can have a degree of sixteen. A polynomial function with a degree of sixteen is a hexadecic function, which means that it has sixteen variable terms.

Q: Can a polynomial function have a degree of seventeen?

A: Yes, a polynomial function can have a degree of seventeen. A polynomial function with a degree of seventeen is a heptadecic function, which means that it has seventeen variable terms.

Q: Can a polynomial function have a degree of eighteen?

A: Yes, a polynomial function can have a degree of eighteen. A polynomial function with a degree of eighteen is an octadecic function, which means that it has eighteen variable terms.

Q: Can a polynomial function have a degree of nineteen?

A: Yes, a polynomial function can have a degree of nineteen. A polynomial function with a degree of nineteen is a nonadecic function, which means that it has nineteen variable terms.

Q: Can a polynomial function have a degree of twenty?

A: Yes, a polynomial function can have a degree of twenty. A polynomial function with a degree of twenty is aicosahoric function, which means that it has twenty variable terms.

Q: Can a polynomial function have a degree of twenty-one?

A: Yes, a polynomial function can have a degree of twenty-one. A polynomial function with a degree of twenty-one is ahenadecic function, which means that it has twenty-one variable terms.

Q: Can a polynomial function have a degree of twenty-two?

A: Yes, a polynomial function can have a degree of twenty-two. A polynomial function with a degree of twenty-two is aicosadecic function, which means that it has twenty-two variable terms.

Q: Can a polynomial function have a degree of twenty-three?

A: Yes, a polynomial function can have a degree of twenty-three. A polynomial function with a degree of twenty-three is ahenadecic function, which means that it has twenty-three variable terms.

Q: Can a polynomial function have a degree of twenty-four?

A: Yes, a polynomial function can have a degree of twenty-four. A polynomial function with a degree of twenty-four is aicosadecic function, which means that it has twenty-four variable terms.

Q: Can a polynomial function have a degree of twenty-five?

A: Yes, a polynomial function can have a degree of twenty-five. A polynomial function with a degree of twenty-five is ahenadecic function, which means that it has twenty-five variable terms.

Q: Can a polynomial function have a degree of twenty-six?

A: Yes, a polynomial function can have a degree of twenty-six. A polynomial function with a degree of twenty-six is aicosadecic function, which means that it has twenty-six variable terms.

Q: Can a polynomial function have a degree of twenty-seven?

A: Yes, a polynomial function can have a degree of twenty-seven. A polynomial function with a degree of twenty-seven is ahenadecic function, which means that it has twenty-seven variable terms.

Q: Can a polynomial function have a degree of twenty-eight?

A: Yes, a polynomial function can have a degree of twenty-eight. A polynomial function with a degree of twenty-eight is aicosadecic function, which means that it has twenty-eight variable terms.

Q: Can a polynomial function have a degree of twenty-nine?

A: Yes, a polynomial function can have a degree of twenty-nine. A polynomial function with a degree of twenty-nine is ahenadec