Math Practice

Polynomial in Standard Form Practice Problems

Standard form means descending powers, but the polynomials that trip people up are the ones that have to be expanded or simplified before they can be ordered. These twelve problems start with a straight reorder and work up to products and differences where the degree only becomes clear after the algebra.

Questions

1.
Write in standard form.
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The exponents are 1, 0, 2. Sorted from highest to lowest they are 2, 1, 0, giving .
2.
Write in standard form.
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The highest power goes first: . Ascending order is a valid way to write a polynomial, but it is not standard form.
3.
Write in standard form. What is its leading coefficient?
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Standard form is . The leading term is , so the leading coefficient is . The sign moves with its term.
4.
What is the degree of once it is written in standard form?
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Standard form is , so the degree is 7. Ordering the terms is exactly what makes the degree visible at the front.
5.
Simplify and write the result in standard form.
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Combine the terms, , and the terms, . The result is , already in descending order.
6.
Simplify and write it in standard form.
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The two terms cancel completely, leaving . What looked like a cubic is a quadratic — you cannot know the degree until like terms are combined.
7.
Express as a polynomial in standard form.
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Expand: . Combine the middle terms: . Expanding two binomials usually lands in standard form on its own once the like terms are merged.
8.
Express as a polynomial in standard form.
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Multiply each term inside by : . The exponents 3, 2, 1 already descend.
9.
Write as a polynomial in standard form.
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Expand: . The terms cancel and the terms cancel, leaving . This is the difference-of-cubes pattern run backwards.
10.
Write in standard form.
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The minus sign flips every term in the bracket: . Combine: . Flipping only the first term inside the bracket is the classic error here.
11.
Write in standard form.
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, so . Subtracting leaves . The cubic terms cancel and the result is quadratic.
12.
After writing in standard form, what is its constant term?
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, so the product is . The constant term is 9 — or, faster, multiply the constants of the three factors: .

Written by

Zohaib

MPhil in Mathematics, Founder of Mathovia

Published

18 September 2026

Mathematics verification: Mathematical formulas, calculations, worked examples and solutions are reviewed for accuracy before publication. How we check our work

Polynomial in Standard Form Practice Problems

The standard form lesson is three steps — expand, simplify, order — and most of the difficulty lives in the first two. These twelve problems start with pure reordering and finish with products where the degree only appears after the cancelling.

The target layout

Exponents strictly descend from left to right. Missing powers are simply left out.

Reordering

Tag each term with its exponent, then sort:

Simplifying before ordering

The cubic terms cancel. Ordering first and reading “degree 3” would be wrong — combine like terms before you decide anything about the polynomial.

Distributing a minus sign

Every term inside the bracket flips sign, not just the first one.

Frequently Asked Questions

What does it mean to write a polynomial in standard form?+

Arrange its terms from the highest power of the variable down to the lowest, with like terms combined. −7x⁶ + x² + 3x + 10 is in standard form; 3x − 7x⁶ + x² + 10 is the same polynomial but not in standard form.

Do I have to simplify before ordering?+

Yes. Combine like terms and expand any brackets first — otherwise you can misread the degree. x³ + 4x − x³ looks cubic but is really just 4x.

What if a power is missing?+

Skip it. x⁵ − 6x + 2 is in standard form even though there are no x⁴, x³ or x² terms. You only write zero placeholders when setting up long or synthetic division.

Does the sign move with the term?+

Always. In 3x − 7x⁶ the second term is −7x⁶, and when it moves to the front the polynomial is −7x⁶ + 3x. Losing that minus sign changes the leading coefficient and the end behavior.

Why is standard form useful?+

The leading term is visible at the front, so the degree and leading coefficient can be read straight off. Long division and synthetic division also rely on the terms being in descending order.

How is standard form different from factored form?+

Standard form is expanded and ordered, like x² − 2x − 15; factored form is a product, like (x + 3)(x − 5). Both describe the same polynomial. Standard form shows the degree; factored form shows the zeros.

x⁵→x⁰Polynomial in Standard Form Practice Problems

Polynomial in Standard Form Practice Problems

The target layout

Reordering

Simplifying before ordering

Distributing a minus sign

Frequently Asked Questions

More practice problems

Polynomial in Standard Form Practice Problems