A polynomial is an algebraic expression that is the sum of multiple monomials, or terms.
Consider the following example polynomial, written in standard form.
4x5+x3−9x−84
polynomialis used more generally.
Just like numbers, polynomials are closed under addition and subtraction. This means, polynomials can be added and subtracted and the result is another polynomial. These operations are performed by combining like terms. Terms with the same variable and exponent are combined by adding or subtracting their coefficients.
Find the sum of the following polynomials. -2x2+x+3and2x2+4x−10
To add these polynomials, like terms will be combined. It can be helpful to rearrange terms so that like terms are next to each other.Calculate the difference between the polynomials. 3x2−4andx2−4x+2
Polynomials are multiplied by using the Distributive Property. For example, for the polynomial product (a+b)(c+d+e), it is possible to distribute (c+d+e) to every term of (a+b). Then, by using the Distributive Property once more, the product can be expressed explicitly. (a+b)(c+d+e)⇓a(c+d+e)+b(c+d+e)⇓ac+ad+ae+bc+bd+be When multiplying two polynomials, each term of the first polynomial multiplies with each term of the second one. This have some important consequences:
Find the product of the polynomials x3+x2+5 and x4+2x+1 by distributing (x4+2x+1) to each term in x3+x2+5. Then, determine the degree of the resulting polynomial. (x3+x2+5)(x4+2x+1)=(x3+x2+5)(x4+2x+1)= x3(x4+2x+1)+x2(x4+2x+1)+ 5 (x4+2x+1) x7+2x4+x3+ x6+2x3+x2+5x4+10x+5 It can be seen that since both polynomials have 3 terms, multiplying them results in 3×3=9 products. Nevertheless, it can be simplified by combining like terms. (x3+x2+5)(x4+2x+1)=(x3+x2+5)(x4+2x+1)= x7+2x4+x3+ x6+2x3+x2+5x4+10x+5 x7+7x4+3x3+ x6+ x2+10x+ 5 It can also be noted that the polynomial x3+x2+5 its of degree 3 and the polynomial x4+2x+1 its of degree 4. When written in standard form, their product is x7+x6+7x4+3x3+x2+10x+5. This is a polynomial of degree 3+4=7.
Find a polynomial that represents the area of the rectangle.