The degree of the polynomial 4x^6 + 5x^2y^2 − 4 is □.

6. 4x^6 has degree 6; 5x^2y^2 has degree 2 + 2 = 4; the constant, 0. A polynomial's degree is its highest term's.

The constant term of the polynomial 2x + 13 + 8x^2 is 2. True or false?

False. The constant term is the one with no letter: 13. 2 is the coefficient of x.

In the polynomial 6x^3 − 9x^2 − x − 8, the coefficient of x^2 is □, the coefficient of x is □ and the constant term is □. Fill in each blank

−9, −1, −8. Each sign goes with the term after it: x^2's coefficient is −9; x alone has coefficient 1 and −x has −1, so here it is −1; the constant is −8.

Polynomials & identities · Ages 13–14

What a polynomial is

A sign goes with the term after it; the degree is the highest term's.

How to do it

Method

A sign goes with the term after it; the degree is the highest term's

4x³ − 2x + 7: three terms, x's coefficient −2, degree 3

Another way

Rewrite it all as adding

4x³ + (−2x) + 7

In full

A monomial is a number, a letter, or a product of numbers and letters: 7, x, −5xy, 3x². Monomials added together make a polynomial, and each monomial is one of its terms. To read 4x³ − 2x + 7, think of each minus as adding a negative: 4x³ + (−2x) + 7. So its three terms are 4x³, −2x and 7, each sign going with the term after it. The number in front of the letters is the coefficient: −2 for −2x, 1 for x alone, −1 for −x. A term with no letter is the constant term. A term's degree is the sum of its letters' indices (3x²y: 2 + 1 = 3); a polynomial's degree is that of its highest term. Written from the highest power down it is in descending powers; from the lowest up, ascending. Anything with a letter in a denominator (such as 1x) is not a polynomial.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
The degree of the polynomial 4x⁶ + 5x²y² − 4 is .
Show the answer and working
Answer: 6
4x⁶ has degree 6; 5x²y² has degree 2 + 2 = 4; the constant, 0. A polynomial's degree is its highest term's.
Example 2
The constant term of the polynomial 2x + 13 + 8x² is 2.
True or false?
Show the answer and working
Answer: False
The constant term is the one with no letter: 13. 2 is the coefficient of x.
Example 3
In the polynomial 6x³ − 9x² − x − 8, the coefficient of x² is , the coefficient of x is and the constant term is .
Fill in each blank
Show the answer and working
Answer: −9, −1, −8
Each sign goes with the term after it: x²'s coefficient is −9; x alone has coefficient 1 and −x has −1, so here it is −1; the constant is −8.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.The degree of the polynomial 5x³ + 5x⁴y² − 15 is .
Answer:
2.The constant term of the polynomial 3x + 5 + 4x² is 5.
True or false?
Answer:
3.In the polynomial 6x² − 8x + 2, what is the coefficient of x?
- A. 8
- B. 6
- C. −8
- D. 2
Choose the right answer
Answer:
4.The degree of the polynomial 2x²y − 5x³y⁴ + 9y⁷ is .
Answer:
5.4x²y² is a monomial of degree 3.
True or false?
Answer:

All the answers

1. 6
2. True
3. C. −8
4. 7
5. False

Free worksheet: What a polynomial isWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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