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Chapter 12: Problem 23

Give the leading coefficient. $$ \sqrt{7} u^{3}+12 u-4+6 u^{2} $$

Short Answer

Expert verified
Answer: The leading coefficient of the given polynomial is $$\sqrt{7}$$.

Step by step solution

01

Identify the term with the highest power of u

In the given polynomial $$ \sqrt{7} u^{3}+12 u-4+6 u^{2} $$, the term with the highest power of u is $$\sqrt{7} u^{3}$$.
02

Find the coefficient of the term with the highest power of u

The coefficient of the term $$\sqrt{7} u^{3}$$ is $$\sqrt{7}$$.
03

Write down the leading coefficient

The leading coefficient of the given polynomial is $$\sqrt{7}$$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Polynomials
A polynomial is a mathematical expression involving a sum of powers in one or more variables, multiplied by coefficients. In simpler terms, it's made up of terms like \( ax^n \), where \( a \) is a number known as the coefficient, \( x \) represents the variable, and \( n \) denotes a non-negative integer exponent.

Polynomials are often seen in forms such as quadratic, cubic, and quartic, depending on their highest power. They are fundamental in algebra, playing a crucial role in expressing equations and functions.
  • Terms are separated by plus or minus signs.
  • Each term consists of a coefficient and a variable raised to a power.

An example of a polynomial could be \( 3x^2 + 2x - 5 \). Each part of this expression is a term, and they combine to form the polynomial.
Importance of the Highest Power Term
In a polynomial, the highest power term is the term with the largest exponent on the variable. Understanding this is important because it often determines the degree of the polynomial, which describes its most significant behavior as the variable grows larger.

For instance, in the polynomial \( \sqrt{7} u^{3}+12 u-4+6 u^{2} \), the term with the highest power is \( \sqrt{7} u^{3} \).
  • The highest power indicates the degree of the polynomial, which in this case is 3, making it a cubic polynomial.
  • This term influences the end behavior of the polynomial graph.
As you identify the highest power term, you also pinpoint the leading term, which is vital for analyzing polynomial functions in calculus and beyond.
The Role of Coefficients
Coefficients are the numerical factors multiplying the variables in a polynomial. They hold significant roles in shaping the polynomial's graph and properties.

In the context of polynomials, each term's coefficient determines how "steep" or "flat" the graph of the polynomial will appear regarding each part. Using the example \( \sqrt{7} u^{3} \), \( \sqrt{7} \) is the coefficient of \( u^3 \).
  • The coefficient affects the amplitude and orientation of the curve.
  • An essential concept is the leading coefficient, located in the term with the highest power.
Understanding coefficients allows you to gain insights into how changes in their values can stretch or shrink polynomial graphs, helping you predict the function's behavior more accurately.

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Most popular questions from this chapter

Refer to the functions \(f(x)\) and \(g(x),\) where the function $$ g(x)=1+\frac{1}{2} x+\frac{3}{8} x^{2}+\frac{5}{16} x^{3} $$ is used to approximate the values of $$ f(x)=\frac{1}{\sqrt{1-x}} $$ Given that $$ f(1 / 2)=\frac{1}{\sqrt{1-\frac{1}{2}}}=\frac{1}{\sqrt{\frac{1}{2}}}=\frac{1}{\frac{1}{\sqrt{2}}}=\sqrt{2} $$ use \(g(x)\) to find a rational number (a fraction) that approximately equals \(\sqrt{2}\).

p(z)=4 z^{3}-z. Find the given values and simplify if possible. The values of \(z\) such that \(p(z)=0\)

Problems \(28-31\) refer to the functions \(f(x)\) and \(g(x),\) where the function $$ g(x)=1+\frac{1}{2} x+\frac{3}{8} x^{2}+\frac{5}{16} x^{3} $$ is used to approximate the values of $$ f(x)=\frac{1}{\sqrt{1-x}} $$ Evaluate \(f\) and \(g\) at \(x=0\). What does this tell you about the graphs of these two functions?

Evaluate the expressions in Problems \(51-54\) given that \(f(x)=2 x^{3}+3 x-3, \quad g(x)=3 x^{2}-2 x-4\) \(h(x)=f(x) g(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}\) $$ n $$

State the given quantities if \(p(x)\) is a polynomial of degree 5 with constant term 3 , and \(q(x)\) is a polynomial of degree 8 with constant term -2. The degree of \(p(x)+q(x)\).
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