Question

identify single-variable polynomials in standard form

which polynomials are in standard form?

(x^2 + 3x + 2)
(q^3 - 15q + 12q^2 - 16)
(4a + a^2 + a - 2)
(3x^4 + 4x^3 - 3x^2 - 1)
(3t^3 + 3t^2 + 2t)
(14 + a^3 - 6a + 8a^2)

Explanation:

🆕 New Concept Discovered: Polynomials in Standard Form
Arranging terms from highest exponent to lowest

Step 1: Define Standard Form

A single-variable polynomial is in standard form when its terms are written in descending order of their exponents (degrees).

This means we look at the power of the variable in each term and make sure they go from highest to lowest:
\[ \text{Highest power}
ightarrow \text{Next highest}
ightarrow \dots
ightarrow \text{Constant term (no variable)} \]

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Step 2: Evaluate Each Option

We will check the exponents of each term from left to right for each given polynomial:

1. \( x^2 + 3x + 2 \)

• Exponents: \( 2

ightarrow 1
ightarrow 0 \)

• This is in descending order. (Standard Form)

1. \( q^3 - 15q + 12q^2 - 16 \)

• Exponents: \( 3

ightarrow 1
ightarrow 2
ightarrow 0 \)

• Not in descending order because \( 1 \) is followed by \( 2 \).

1. \( 4a + a^2 + a - 2 \)

• Exponents: \( 1

ightarrow 2
ightarrow 1
ightarrow 0 \)

• Not in descending order because \( 1 \) is followed by \( 2 \).

1. \( 3x^4 + 4x^3 - 3x^2 - 1 \)

• Exponents: \( 4

ightarrow 3
ightarrow 2
ightarrow 0 \)

• This is in descending order. (Standard Form)

1. \( 3t^3 + 3t^2 + 2t \)

• Exponents: \( 3

ightarrow 2
ightarrow 1 \)

• This is in descending order. (Standard Form)

1. \( 14 + a^3 - 6a + 8a^2 \)

• Exponents: \( 0

ightarrow 3
ightarrow 1
ightarrow 2 \)

• Not in descending order.

Answer:

The polynomials in standard form are:

• \( x^2 + 3x + 2 \)
• \( 3x^4 + 4x^3 - 3x^2 - 1 \)
• \( 3t^3 + 3t^2 + 2t \)