Simplifying Polynomials: Step-by-Step Guide
A polynomial is an expression made of terms like 3x^2, 5x, and 7, added or subtracted together. To simplify a polynomial, remove any parentheses, combine the like terms (terms with the same letter and the same exponent), and write the answer with the highest power first.
Math Racer skill guide · Algebra 1 · Grade 9
How to solve it, step by step
- Remove the parentheses. Multiply any number outside the parentheses by each term inside. If there is a minus sign in front of the parentheses, flip the sign of every term inside.\[k(ax^{n}+bx^{m}) = kax^{n}+kbx^{m}\]
- Find the like terms. Like terms have the same letter and the same exponent. 3x^2 and 5x^2 are like terms. 3x^2 and 5x are not.\[ax^{n}\text{ and }bx^{n}\text{ are like terms}\]
- Combine the like terms. Add or subtract the numbers in front. Keep the letter and the exponent the same.\[ax^{n}+bx^{n} = (a+b)x^{n}\]
- Write the answer in standard form. Put the term with the highest exponent first, then the next highest, and so on. Leave out any term that is zero.\[a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}\]
Worked example
Simplify: (3x^2 + 5x - 4) - (x^2 - 2x + 6)
Step 1: Distribute the minus sign to every term in the second parentheses: 3x^2 + 5x - 4 - x^2 + 2x - 6.
Step 2: Find the like terms: 3x^2 and -x^2, then 5x and 2x, then -4 and -6.
Step 3: Combine them: 3x^2 - x^2 = 2x^2, 5x + 2x = 7x, and -4 - 6 = -10.
Step 4: Write in order from the highest power: 2x^2 + 7x - 10.
Check: Try x = 2 in the original: (12 + 10 - 4) - (4 - 4 + 6) = 18 - 6 = 12. Try x = 2 in the answer: 8 + 14 - 10 = 12. They match.
Tips and common mistakes
- When you subtract a polynomial, flip the sign of EVERY term inside the parentheses.
- x^2 and x are not like terms, because the exponents are different.
- Be careful with signs. -3x^2 + 5x^2 = 2x^2, not 8x^2.
- Write the final answer with the highest power first.
- If a term ends up with 0 in front, leave it out.
- Check your work by trying a number for x in both the original and the answer.
- When you add like terms, the exponent does not change. x^2 + x^2 = 2x^2, not x^4.