How to Write Polynomials in Standard Form? (+FREE Worksheet!)
When working with polynomials, you should always write them in standard form.
Tutor-style math help
Write Polynomials in Standard Form: what to notice and how to work it
Polynomials skill
Polynomial problems reward structure. Before expanding, look for degree, leading term, common factors, and familiar products.
What to notice first
Put the polynomial in standard form when possible. The leading term tells end behavior, and factors reveal zeros.
Common student mistake
Do not cancel or combine unlike terms. \(x^2\), \(x\), and constants are different kinds of terms.
Key formulas and cues
\(a^2-b^2=(a-b)(a+b)\)
\((a+b)^2=a^2+2ab+b^2\)
\(P(c)=0\Rightarrow (x-c)\text{ is a factor}\)
A reliable path
- Organize by degreeWrite terms from highest power to lowest power.
- Look for structureTry GCF, special products, grouping, or division depending on the expression.
- Check with featuresZeros, multiplicity, and end behavior should agree with your algebra.
Worked examples
Combine like terms
Example: \(3x^2+5x-x^2+2x\)
- Group x squared terms.
- Group x terms.
- Combine each group.
Answer: \(2x^2+7x\)
Factor a difference of squares
Example: \(x^2-25\)
- Recognize a squared term minus a squared term.
- Use a^2 – b^2.
- Write conjugate factors.
Answer: \((x-5)(x+5)\)
Try one before moving on
Try: Factor \(x^2+7x+12\).
Answer: \((x+3)(x+4)\).
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
x
Write Polynomials in Standard Form: pop-up practice
Answer these quick questions, then use the feedback to decide which part of the lesson to review.
Choose an answer to begin.
1. \(2x+3x=\)
2. \(x^2-16=\)
3. The degree comes from the:
Watch this practice video for additional examples and reinforcement:
Related Topics
- How to Multiply Monomials
- How to Multiply and Dividing Monomials
- How to Multiply Binomials
- How to Factor Trinomials
- How to Add and Subtract Polynomials
Step by step guide to writing polynomials in standard form
- A polynomial function \(f(x)\) of degree \(n\) is of the form
\(f(x)=a_{n} x^{n}+a_{n-1} x_{n-1}+⋯+ a_{1} x+a_{0}\) - The first term is the one with the biggest power!
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Writing Polynomials in Standard Form – Example 1:
Write this polynomial in standard form. \(8+5x^2-3x^3=\)
Solution:
The first term is the one with the biggest power: \(8+5x^2−3x^3=−3x^3+5x^2+8\)
Writing Polynomials in Standard Form – Example 2:
Write this polynomial in standard form. \(5x^2−9x^5+8x^3−11= \)
Solution:
The first term is the one with the biggest power: \(5x^2−9x^5+8x^3−11=
−9x^5+8x^3+5x^2−11 \)
Writing Polynomials in Standard Form – Example 3:
Write this polynomial in standard form. \(-12+3x^2-6x^4=\)
Solution:
The first term is the one with the biggest power: \(-12+3x^2-6x^4=-6x^4+3x^2-12\)
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Exercises
Write each polynomial in standard form.
- \(\color{blue}{3x^2 – 5x^3}\)
- \(\color{blue}{3 + 4x^3 – 3}\)
- \(\color{blue}{2x^2 + x – 6x^3}\)
- \(\color{blue}{9x^4 – 7x^6}\)
- \(\color{blue}{12 – 7x + 9x^4}\)
- \(\color{blue}{5x^2 + 13x – 2x^3}\)
Download Writing Polynomials in Standard Form Worksheet
- \(\color{blue}{- 5x^3 + 3x^2}\)
- \(\color{blue}{4x^3}\)
- \(\color{blue}{-6x^3 + 2x^2 + x}\)
- \(\color{blue}{ – 7x^6+9x^4}\)
- \(\color{blue}{9x^4 – 7x + 12}\)
- \(\color{blue}{-2x^3 + 5x^2 + 13x}\)
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