A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
The degree of a polynomial is the highest power of the variable in the expression.
Polynomial Arithmetic
Multiplying a Polynomial by a Constant
To multiply a polynomial by a constant, multiply each term of the polynomial by the constant.
Multiply $3x^2 - 2x + 5$ by $4$:
\$\$4 \cdot (3x^2 - 2x + 5) = 12x^2 - 8x + 20\$\$
Adding and Subtracting Polynomials
To add or subtract polynomials, combine the like terms (terms with the same variable and exponent).
Add $(x^2 - 5x + 6)$ and $(2x^2 + 3x - 2)$:
- Combine $x^2$ terms: $x^2 + 2x^2 = 3x^2$
- Combine $x$ terms: $-5x + 3x = -2x$
- Combine constant terms: $6 - 2 = 4$
The result is $3x^2 - 2x + 4$.
Subtract $(x^2 - 2x + 3)$ from $(3x^2 + 5x - 1)$:
- Distribute the negative sign: $3x^2 + 5x - 1 - (x^2 - 2x + 3)$
- Combine like terms: $3x^2 - x^2 + 5x + 2x - 1 - 3$
The result is $2x^2 + 7x - 4$.
Forgetting to distribute the negative sign when subtracting polynomials is a common error.
Multiplying Polynomials
To multiply polynomials, multiply each term in the first polynomial by each term in the second polynomial, then combine like terms.
Multiply $(x + 3)(2x^2 - 4x + 5)$:
- Multiply $x$ by each term in $2x^2 - 4x + 5$: $x \cdot 2x^2 = 2x^3$, $x \cdot (-4x) = -4x^2$, $x \cdot 5 = 5x$
- Multiply $3$ by each term in $2x^2 - 4x + 5$: $3 \cdot 2x^2 = 6x^2$, $3 \cdot (-4x) = -12x$, $3 \cdot 5 = 15$
- Combine like terms: $2x^3 + (-4x^2 + 6x^2) + (5x - 12x) + 15$
The result is $2x^3 + 2x^2 - 7x + 15$.
For binomials, the FOIL method (First, Outer, Inner, Last) can be a helpful mnemonic.
Multiplication Patterns
Perfect Square Trinomials
The square of a binomial $(x + a)^2$ expands to $x^2 + 2ax + a^2$.
Simplify $(x - 3)^2$:
- The coefficient of $x$ is $2 \cdot (-3) = -6$
- The constant term is $(-3)^2 = 9$
The result is $x^2 - 6x + 9$.
Difference of Squares
The product of conjugate binomials $(x - a)(x + a)$ is $x^2 - a^2$.
Simplify $(x - 3)(x + 3)$:
The result is $x^2 - 9$.
Dividing Polynomials
Long Division
Polynomial long division is similar to numerical long division.
Divide $2x^3 + x^2 - 11x + 12$ by $x + 3$:
- Divide the first term of the dividend by the first term of the divisor: $\frac{2x^3}{x} = 2x^2$
- Multiply the entire divisor by $2x^2$ and subtract from the dividend
- Repeat the process with the new polynomial until the degree of the remainder is less than the degree of the divisor
The quotient is $2x^2 - 5x + 4$ with a remainder of $0$.
The remainder can be expressed as a fraction: $\frac{\text{remainder}}{\text{divisor}}$.
- Multiply $(2x - 3)(x^2 + 4x + 5)$ and simplify.
- Divide $3x^3 - 2x^2 + 4x - 5$ by $x - 1$ using long division.
How do the patterns in polynomial arithmetic reflect the underlying structure of mathematics? Are these patterns discovered or invented?
Polynomial Arithmetic in Computer Graphics
Polynomials are used in computer graphics to model curves and surfaces. For example, Bézier curves, which are widely used in vector graphics and animation, are defined using polynomial equations. Understanding polynomial arithmetic is essential for manipulating these curves to create smooth and realistic animations.
Be careful with signs when adding, subtracting, or multiplying polynomials. A small mistake can lead to incorrect results.
The difference of squares identity can be proven by expanding the product:
\$\$ (x - a)(x + a) = x^2 + ax - ax - a^2 = x^2 - a^2 \$\$
To divide polynomials, you can also use synthetic division, a more efficient method for divisors of the form $x - c$. However, this method is not covered in this section.
