Polynomial Review · Lesson 1
Before you can operate on polynomials you have to read them: to look at a jumble of terms and instantly see how many terms there are, which powers appear, and which term is in charge. Degree is the single most important label, because the highest-degree term controls how the whole expression behaves.
By the end, you can
A term is a single product of a number and variable powers: 7x³, −2x, and 5 are all terms. The number multiplying the variable is the coefficient; in −2x the coefficient is −2. A term with no variable, like 5, is the constant term. A polynomial is terms joined by + or −.
Why it works
The +/− signs are the seams that stitch terms together. Read 3x² − 5x + 4 as (+3x²) + (−5x) + (+4): a term always carries its own sign. That one habit makes subtraction painless later.
The degree of a term is the exponent on its variable. 7x³ has degree 3; 4x has degree 1 (the invisible exponent is 1); a nonzero constant like 5 has degree 0, because 5 = 5x⁰. The coefficient in front never changes the degree — only the exponent does, so −11x⁴ still has degree 4.
Why it works
Degree measures how fast a term grows when x gets big. A degree-3 term eventually dwarfs a degree-2 term, which dwarfs a degree-1 term. That growth ranking is exactly why we sort by degree.
The degree of a polynomial is the largest degree among its terms. The term owning that degree is the leading term, and its coefficient is the leading coefficient. In −2x⁴ + 9x² − x + 3 the degree is 4, the leading term is −2x⁴, and the leading coefficient is −2.
Why it works
When x is huge, the highest-degree term drowns out all the others, so it controls the polynomial's long-run personality. That is why it gets top billing — it literally leads.
Count the terms AFTER combining like terms: 1 term = monomial (5x²), 2 terms = binomial (x + 7), 3 terms = trinomial (x² − 3x + 2). Four or more just get called a polynomial. Term count and degree are independent: x² − 3x + 2 is a degree-2 trinomial, and 4x³ is a degree-3 monomial.
Why it works
Count the terms, then measure the tallest. Term count is a headcount; degree is a height. A trinomial is not automatically degree 3.
Write terms from highest degree to lowest: 5 − 2x³ + x becomes −2x³ + x + 5. Standard form is not required for correctness, but it makes the degree, leading coefficient, and constant jump out and lines terms up for adding and subtracting.
Why it works
It is alphabetizing for math. Once everything is sorted tallest-first, the leading term is simply the one on the left and the constant is the one on the right.
Classify 4x² − 7. Give its degree, leading coefficient, and name.
Count the terms
4x² and −7 → two terms → binomial.
Find the highest degree
The exponents are 2 and 0, so the degree is 2.
Read the leading parts
Leading coefficient is 4; the constant is −7.
State the degree and the coefficient of the term −6x⁴.
Read the exponent for the degree
The exponent on x is 4, so the degree is 4.
Read the number in front
The coefficient is −6 (the minus is part of it) and does not affect the degree.
Put 5 − 2x³ + x in standard form and name it.
Rank by degree
−2x³ (deg 3), x (deg 1), 5 (deg 0).
Write descending
−2x³ + x + 5.
Name it
Three terms → trinomial; degree 3 (cubic); leading coefficient −2.
What is the degree of the constant polynomial 9? What about 0?
Rewrite 9
9 = 9x⁰, a nonzero term of degree 0, so the degree is 0.
Handle 0 separately
0 has no nonzero term at all, so there is nothing to measure — its degree is undefined.
After combining like terms, name and give the degree of 3x² + 5x − x² − 5x + 8.
Combine the x² terms
3x² − x² = 2x².
Combine the x terms
5x − 5x = 0, so they vanish.
Read the result
2x² + 8 → two terms → binomial, degree 2.
It has an x, so the degree is at least 1.
A constant has degree 0 (5 = 5x⁰). Don't invent a degree from a lone number, and don't assume every variable term is degree 1 — check the exponent.
Counting terms before combining like terms.
Always simplify first. 3x² + 5x − x² − 5x + 8 looks like five terms but is really the binomial 2x² + 8. Classify the true object, not the raw expression.
The degree is the exponent on whichever term is written first.
Degree is the maximum term-degree, not the leftmost. Sort to standard form first, or scan every term for the highest exponent.
The polynomial 0 has degree 0.
0 has NO degree (undefined) because it has no nonzero leading term to measure. A nonzero constant like 7 does have degree 0, since 7 = 7x⁰.
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t₁ = 5, d = 3 and t1=5,d=3 both count. For a list of numbers, separate them with commas (e.g. 3, 10, 17).How many terms does 3x² − 5x + 4 have, and what is it called?
What is the degree of the term 7x³? (Enter a number.)
What is the degree of the term −11x⁴? (Enter a number.)
What is the leading coefficient of −2x⁴ + 9x² − x + 3?
What is the degree of the constant term 5? Remember 5 = 5x⁰. (Enter a number.)
Which is 5 − 2x³ + x written in standard (descending) form?
Simplify first, then classify: 3x² + 5x − x² − 5x + 8.
Which statement is true?
What is the degree of the polynomial 0?
State the leading coefficient of −x³ + 6x² − 9. (Enter a number.)
x⁵ − x³ + 1 is best described as:
The polynomial ax³ + 4x² − x + a is said to have degree 2. What is a, and what is the resulting polynomial?
Why does 0 have no degree while 7 has degree 0?
Combine like terms in 2x² − 2x² + 7x, then enter the degree of the result. (Enter a number.)