Math  /  Algebra

QuestionWhich term in the polynomial 2x3+5x23x+12 x^{3}+5 x^{2}-3 x+1 has the leading coefficient? (A) 1
B 3x-3 x C 5x25 x^{2} (D) 2x32 x^{3}

Studdy Solution

STEP 1

What is this asking? Which of these chunks in the polynomial has the coefficient attached to the highest power of xx? Watch out! Don't get tricked by the order they appear in; we're looking for the highest power of xx!

STEP 2

1. Identify the terms.
2. Find the degrees.
3. Spot the highest degree.
4. Find the leading coefficient.

STEP 3

Let's **carefully** look at our polynomial: 2x3+5x23x+12x^3 + 5x^2 - 3x + 1.
Remember, a *term* in a polynomial is a product of a number (the coefficient) and a variable raised to a power.

STEP 4

Our terms are 2x32x^3, 5x25x^2, 3x-3x, and 11.
Notice how we kept the sign with the term!
That's super important!

STEP 5

The *degree* of a term is simply the power of the variable.
So, the degree of 2x32x^3 is **3**.
The degree of 5x25x^2 is **2**.
The degree of 3x-3x is **1** (because 3x-3x is the same as 3x1-3x^1).

STEP 6

What about the degree of 11?
Well, 11 can be written as 1x01x^0 (remember anything to the power of zero is one!), so its degree is **0**.

STEP 7

Out of all our degrees (3, 2, 1, and 0), the **highest** degree is **3**.

STEP 8

The *leading coefficient* is the coefficient of the term with the highest degree.
The term with the highest degree is 2x32x^3.

STEP 9

The coefficient of 2x32x^3 is **2**.
So, our **leading coefficient** is **2**!
And the whole term is 2x32x^3.

STEP 10

The term with the leading coefficient is 2x32x^3, so the answer is (D).

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