Question

The leading coefficient of the polynomial \(3 x^{2}-8 x+4\) is \(\underline{?}\)

Short answer

The leading coefficient of the polynomial \(3 x^{2}-8 x+4\) is 3.

Step-by-step solution

Identify the Polynomial

The given polynomial is \(3 x^{2}-8 x+4\).

Identify the Term with the Highest Degree

The term with the highest degree in the polynomial is \(3 x^{2}\).

Identify the Leading Coefficient

The leading coefficient is the numerical factor before the term with the highest degree. In this case, that is 3.

Key concepts

Polynomials

When we talk about polynomials, we're referring to algebraic expressions that consist of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents. Think of them as a string of mathematical terms joined together like beads on a necklace.

For example, the expression provided in the exercise, \(3x^2 - 8x + 4\), is a polynomial. It's made up of three terms: \(3x^2\), \(-8x\), and \(4\). What makes it a polynomial is the smooth blend of these terms to express a single, coherent mathematical idea.

Coefficient

What exactly is a coefficient? In simplest terms, it is the number that multiplies a variable within an algebraic expression, like a silent partner giving strength to the variable. In our exercise, in each term of the polynomial \(3x^2 - 8x + 4\), the coefficients are 3, -8, and 4, respectively.

It's helpful to think about a coefficient as a 'scaling factor' for the variable it accompanies. When we say the leading coefficient, we're highlighting the coefficient attached to the first term when the polynomial is written in standard form, with terms from highest to lowest exponent.

Degrees of a Polynomial

The concept of the degree of a polynomial is akin to describing the 'level' or 'height' of its elements. It's the highest power of the variable in a polynomial. Looking at our example, \(3x^2 - 8x + 4\), the degree of this polynomial is 2 because the highest exponent of \(x\) present is 2.

The degree gives us a sneak peek into the behavior of the polynomial, particularly its growth rate and the number of roots or solutions it's likely to have. Undoubtedly, understanding the degree is a key step in unraveling the full story of a polynomial.

Algebraic Expressions

Algebraic expressions are like sentences written in the language of mathematics, conveying ideas using numbers, variables, and operations. They form the foundational blocks of algebra and help us describe quantities and relationships.

The expression in our exercise, \(3x^2 - 8x + 4\), is a static snapshot capturing a relationship between \(x\) and its respective outputs. Unlike an equation, an expression doesn't have an equality sign; it's not trying to state that two things are the same, but rather it's presenting a quantity or, in our case, a polynomial.