Polynomial Operations
Grade 10 · Math · Study Guide
Adding, subtracting, multiplying, and dividing polynomials with confidence.
A polynomial is an algebraic expression made up of terms with variables raised to non-negative integer exponents. Terms are separated by addition or subtraction. The degree of a polynomial is the highest exponent present. For example, 3x⁴ - 2x² + 7 is a degree-4 polynomial with three terms (a trinomial).
Adding and subtracting polynomials is straightforward: combine like terms — terms with the same variable and the same exponent. For example, (3x² + 2x - 1) + (x² - 5x + 4) = 4x² - 3x + 3. When subtracting, distribute the negative sign carefully before combining.
Multiplying polynomials uses the distributive property. Every term in the first polynomial multiplies every term in the second. Multiplying two binomials is often done with FOIL (First, Outer, Inner, Last): (x + 3)(x - 2) = x² - 2x + 3x - 6 = x² + x - 6. For longer polynomials, organize work vertically or horizontally to avoid missing terms.
Dividing polynomials can be done with long division, similar to arithmetic long division, or with synthetic division when dividing by a linear factor (x - c). The Remainder Theorem states that when you divide polynomial p(x) by (x - c), the remainder equals p(c). This makes checking divisibility fast: if p(c) = 0, then (x - c) is a factor of p(x).
Key terms
- Polynomial
- An expression with variables raised to non-negative integer exponents, combined by addition/subtraction.
- Degree
- The highest exponent in a polynomial. Determines its classification and end behavior.
- Like Terms
- Terms with the same variable raised to the same power; can be combined by adding coefficients.
- FOIL
- A method for multiplying two binomials: First, Outer, Inner, Last products are added.
- Remainder Theorem
- When p(x) is divided by (x - c), the remainder equals p(c).
Flashcards
- Polynomial
- An expression with terms of the form axⁿ where n is a non-negative integer
- Degree of a polynomial
- The highest exponent among all terms
- FOIL method
- Multiply binomials: (a+b)(c+d) = ac + ad + bc + bd
- Remainder Theorem
- Dividing p(x) by (x-c) gives remainder p(c)
- Like terms
- Terms with identical variable parts; combine by adding coefficients
- Factor Theorem
- If p(c)=0, then (x-c) is a factor of p(x)
Practice questions
What is the degree of the polynomial 5x³ - 2x⁵ + 4x - 7?
- 3
- 4
- 5
- 7
Show answer
Answer: 5
The degree is the highest exponent. The term -2x⁵ has exponent 5, which is the largest.
(2x² + 3x - 1) + (x² - x + 4) = ___x² + 2x + 3.
Show answer
Answer: 3
Add like terms: 2x²+x²=3x². So the coefficient of x² is 3.
When subtracting polynomials, you must distribute the negative sign to every term in the second polynomial.
Show answer
Answer: True
Subtraction means adding the opposite: -(ax²+bx+c) = -ax²-bx-c. Every term changes sign.
Using FOIL, what is (x + 4)(x - 3)?
- x² - 7x - 12
- x² + x - 12
- x² + 7x - 12
- x² - x - 12
Show answer
Answer: x² + x - 12
FOIL: x·x + x·(-3) + 4·x + 4·(-3) = x² - 3x + 4x - 12 = x² + x - 12.
According to the Remainder Theorem, if p(2) = 0, then (x - 2) is a factor of p(x).
Show answer
Answer: True
The Remainder Theorem says the remainder when dividing by (x-c) is p(c). If p(c)=0, the remainder is 0, meaning (x-c) divides evenly — it's a factor.
Fun facts
- The word 'polynomial' comes from the Greek 'poly' (many) and Latin 'nomen' (name/term) — so it literally means 'many terms.'
- Polynomials are used in computer graphics to draw smooth curves — the Bezier curves in every font and vector graphic are built from polynomial equations.
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