Operations with polynomials

A polynomial in x is an expression of the form Polynomials A polynomial in x is an expression of the form axn + bxn–1 + cxn–2 + ... + px2 + qx + r where a, b, c, … are constant coefficients and n is a nonnegative integer. a is called the leading coefficient. Examples of polynomials include: Teacher notes Explain that, in a polynomial, x can only by raised to the power of a positive whole number. Draw the students’ attention to the way the polynomials are written. Point out that they are usually written in descending powers of x, but they can be written in ascending powers of x, especially when the leading coefficient is negative. Mathematical Practices 6) Attend to precision. Students should know what is meant by the terms “polynomial” and “coefficient” and use these in their discussions and written work. 3x7 + 4x3 – x + 8 x11 – 2x8 + 9x and 5 + 3x2 – 2x3.

Polynomials The degree, or order, of a polynomial is given by the highest power of the variable. A polynomial of degree 1 is called linear and has the general form ax + b. A polynomial of degree 2 is called quadratic and has the general form ax2 + bx + c. A polynomial of degree 3 is called cubic and has the general form ax3 + bx2 + cx + d. Teacher notes Point out that any of the coefficients (except a) or the constant term could be 0 and so each expression could involve fewer terms than those shown. Mathematical Practices 6) Attend to precision. Students should know what is meant by the terms “degree”, “order”, “linear”, “quadratic”, “cubic”, and “quartic” and use these terms in their discussions and written work. A polynomial of degree 4 is called quartic and has the general form ax4 + bx3 + cx2 + dx + e.

Using function notation Polynomials are often expressed using function notation. For example, consider the polynomial function: f(x) = 2x2 – 7 We can use this notation to substitute given values of x. Find f(x) when a) x = –2 b) x = t + 1 a) f(–2) = 2(–2)2 – 7 b) f(t + 1) = 2(t + 1)2 – 7 = 8 – 7 = 2(t2 + 2t + 1) – 7 = 1 = 2t2 + 4t + 2 – 7 = 2t2 + 4t – 5

Adding and subtracting polynomials When two or more polynomials are added, subtracted or multiplied, the result is another polynomial. Polynomials are added and subtracted by combining like terms. For example: f(x) = 2x2 – 5x + 4 and g(x) = 2x – 4 Find: a) f(x) + g(x) b) f(x) – g(x) Teacher notes Establish that f(x) is an example of a polynomial of degree 2 and g(x) is a polynomial of degree 1. Remind students that when a polynomial is subtracted all of its terms change sign. Sometimes the result when two polynomials are added, subtracted or multiplied is a constant or 0. We can think of a constant as a polynomial of degree 0 since it can be written in the form ax0. A result, 0 is a special case and is sometimes called the zero polynomial. a) f(x) + g(x) b) f(x) – g(x) = 2x2 – 5x + 4 + 2x – 4 = 2x2 – 5x + 4 – (2x – 4) = 2x2 – 3x = 2x2 – 5x + 4 – 2x + 4 = 2x2 – 7x + 8

Multiplying polynomials When two polynomials are multiplied together, every term in the first polynomial must by distributed to every term in the second polynomial. The distributive property is used to rewrite the expression without parentheses. For example: f(x) = 3x2 – 2 and g(x) = x2 + 5x – 1 f(x) g(x) = (3x2 – 2)(x2 + 5x – 1) Teacher notes Remind students that when two terms with the same base are multiplied together, their exponents are added. Orange lines are used in this example to show each term in the first parentheses multiplying each term in the second parentheses. With practice, this step can be carried out mentally. We could also write 3x 2(x 2 + 5x – 1) – 2(x 2 + 5x – 1) as an intermediary step. Note that multiplying two terms in the first parentheses by three terms in the second parentheses will result in six terms to be combined together, and 2 × 3 = 6. Similarly, if both polynomial functions had three terms, the result would be nine terms to be combined together. = 3x4 + 15x3 – 3x2 – 2x2 – 10x + 2 = 3x4 + 15x3 – 5x2 – 10x + 2