Standard Form • Degree • Leading Coefficient • Classification
Standard Form
A polynomial is in standard form when its terms are written in descending order of degree (highest power first).
Example: \( 3x – 5 + 2x^2 \) → \( 2x^2 + 3x – 5 \)
Degree of a Polynomial
The degree is the highest power of the variable that appears after the polynomial is written in standard form.
Example: \( 5x^3 – 2x + 7 \) has degree 3.
Leading Term & Leading Coefficient
The leading term is the term with the highest degree. The leading coefficient is the coefficient of the leading term.
Example: \(-x^4+7x^6\)
Standard form: \(7x^6-x^4\)
Leading term: \(7x^6\)
Leading coefficient: \(7\)
Classifying by Degree
Constant (degree 0) Linear (degree 1) Quadratic (degree 2) Cubic (degree 3) Quartic (degree 4) Quintic (degree 5)
Classifying by Number of Terms
Monomial – 1 term Binomial – 2 terms Trinomial – 3 terms Polynomial – 4 or more terms
Tip: Always rewrite the polynomial in standard form first. Then identify the degree, leading term, leading coefficient, and classification.
Write each polynomial in standard form (descending powers of the variable).
Write the polynomial in standard form: \( 4x - 1 + x^2 \)
Arrange the terms in descending order of degree:
$$x^2 + 4x - 1$$
Answer: \( x^2 + 4x - 1 \)
Write the polynomial in standard form: \( -4x^3 + x^5 – 2x + 7x^2 \)
Arrange the terms from highest degree to lowest:
$$x^5 – 4x^3 + 7x^2 – 2x$$
Answer: \( x^5 – 4x^3 + 7x^2 – 2x \)
Write in standard form and state the degree: \( 8 – 3x^4 + 5x – x^2 + 2x^3 \)
Standard form (descending powers):
$$-3x^4 + 2x^3 – x^2 + 5x + 8$$
The highest power is 4, so the degree is 4.
Answer: \( -3x^4 + 2x^3 – x^2 + 5x + 8 \) (degree 4)
Identify the degree of each polynomial.
What is the degree of the polynomial \( 4x^3 + 5x - 2 \)?
The highest power of \( x \) is 3.
Answer: 3
What is the degree of the polynomial \( -4x^2 + 9x^5 – x + 1 \)?
Rewrite in standard form: \( 9x^5 – 4x^2 – x + 1 \)
The highest power is 5.
Answer: 5
What is the degree of the polynomial \( 7 – 3x^4 + 2x(x^3 – 5) \)?
First expand:
$$2x(x^3 – 5) = 2x^4 – 10x$$
Substitute back:
$$7 – 3x^4 + 2x^4 – 10x = -x^4 – 10x + 7$$
Highest power is 4.
Answer: 4
Identify the leading coefficient of each polynomial.
What is the leading coefficient of the polynomial \( 2x^3 - 5x^7 + 4 \)?
Rewrite in standard form: \(-5x^7 + 2x^3 + 4\)
The leading term is \(-5x^7\).
The leading coefficient is \(-5\).
Answer: \(-5\)
What is the leading coefficient of the polynomial \( 5 – 3x^2 + 8x^7 – x \)?
Rewrite in standard form: \( 8x^7 – 3x^2 – x + 5 \)
The leading term is \( 8x^7 \).
The leading coefficient is 8.
Answer: 8
What is the leading coefficient of \( (2x – 1)(x^3 + 4) – 3x^4 \)?
First expand \( (2x – 1)(x^3 + 4) \):
$$2x \cdot x^3 = 2x^4$$
$$2x \cdot 4 = 8x$$
$$-1 \cdot x^3 = -x^3$$
$$-1 \cdot 4 = -4$$
So \( 2x^4 – x^3 + 8x – 4 \)
Subtract \( 3x^4 \):
$$2x^4 – x^3 + 8x – 4 – 3x^4 = -x^4 – x^3 + 8x – 4$$
Leading term is \( -x^4 \), so leading coefficient is −1.
Answer: −1
Classify each polynomial by its degree (constant, linear, quadratic, cubic, etc.).
Classify the polynomial \( 4x^2 – 9 \) by degree.
The highest power is 2, so it is quadratic.
Answer: Quadratic
Classify the polynomial \( -7x^5 + 2x^3 – x \) by degree.
The highest power is 5, so it is quintic (5th-degree).
Answer: Quintic (or 5th-degree)
Classify by degree after simplifying: \( 3x(x^2 – 4) + 5x^3 \)
Expand:
$$3x(x^2 – 4) = 3x^3 – 12x$$
Add \( 5x^3 \):
$$3x^3 – 12x + 5x^3 = 8x^3 – 12x$$
Highest power is 3 → cubic.
Answer: Cubic
Classify each polynomial by the number of terms (monomial, binomial, trinomial, or polynomial).
Classify the polynomial \( 5x^3 \) by number of terms.
There is only one term.
Answer: Monomial
Classify the polynomial \( 2x^2 – 7x + 4 \) by number of terms.
There are three terms.
Answer: Trinomial
After simplifying \( (x + 2)(x – 2) + 5x \), classify the result by number of terms.
First expand:
$$(x + 2)(x – 2) = x^2 – 4$$
Add \( 5x \):
$$x^2 – 4 + 5x = x^2 + 5x – 4$$
There are three terms.
Answer: Trinomial
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