Sect. 5.1 Polynomials and Polynomial Functions Definitions Terms Degree of terms and polynomials Polynomial Functions Evaluating Graphing Simplifying by Combining Like Terms Adding & Subtracting Polynomials 5.1

Definitions An algebraic term is a number or a product of a number and a variable (or variables) raised to a positive power. Examples: 7x or -11xy2 or 192 or z A constant term contains only a number A variable term contains at least one variable and has a numeric part and a variable part A polynomial expression is: one or more terms separated by addition or subtraction; any exponents must be whole numbers; no variable in any denominator. Monomial – one term: -47 or 7x or 92xyz5 Binomial – two terms: a + b or 7x5 – 44 Trinomial – three terms: x2 + 6x + 9 or a + b – 55c 5.1

Ordering a Polynomial’s Terms If there are multiple variables, one must be specified Descending Order – Variable with largest exponent leads off (this is the Standard) Ascending Order – Constant term leads off Arrange in ascending order: Arrange in descending order of x: 5.1

Of a Monomial: (a Single Term) DEGREE Of a Monomial: (a Single Term) Several variables – the degree is the sum of their exponents One variable – its degree is the variable’s exponent Non-zero constant – the degree is 0 The constant 0 has an undefined degree Of a Polynomial: (2 or more terms added or subtracted) Its degree is the same as the degree of the term in the polynomial with largest degree (leading term?). 5.1

Polynomial Functions Equations in one variable: f(x) = 2x – 3 (straight line) g(x) = x2 – 5x – 6 (parabola) h(x) = 3x3 + 4x2 – 2x + 5 Evaluate by substitution: f(5) = 2(5) – 3 = 10 – 3 = 7 g(-2) = (-2)2 – 5(-2) – 6 = 4 + 10 – 6 = 8 h(-1) = 3(-1)3 + 4(-1)2 – 2(-1) + 5 = -3+ 4+2+5= 8 5.1

Opposites of Monomials The opposite of a monomial has a different sign The opposite of 36 is -36 The opposite of -4x2 is 4x2 Monomial: Opposite: -2 2 5y -5y ¾y5 -¾y5 -x3 x3 0 0 5.1

Writing Any Polynomial as a Sum -5x2 – x is the same as -5x2 + (-x) Replace subtraction with addition: Keep the negative sign with the monomial 4x5 – 2x6 – 4x + 7 is 4x5 + (-2x6) + (-4x) + 7 You try it: -y4 + 3y3 – 11y2 – 129 -y4 + 3y3 + (-11y2) + (-129) 5.1

Identifying Like Terms When several terms in a polynomial have the same variable(s) raised to the same power(s), we call them like terms. 3x + y – x – 4y + 6x2 – 2x + 11xy Like terms: 3x, -x, -2x Also: y, -4y You try: 6x2 – 2x2 – 3 + x2 – 11 Like terms: 6x2, -2x2, x2 Also: -3, -11 5.1

Collecting Like Terms (simplifying) The numeric factor in a term is its coefficient. 3x + y – x – 4y + 6x2 – 2x 3 1 -1 -4 6 -2 You can simplify a polynomial by collecting like terms, summing their coefficients Let’s try: 6x2 – 2x2 – 3 + x2 – 11 Sum of: 6x2 + -2x2 + x2 is 5x2 Sum of: -3 + -11 is -14 Simplified polynomial is: 5x2 – 14 5.1

Collection Practice 2x3 – 6x3 = -4x3 5x2 + 7 + 4x4 + 2x2 – 11 – 2x4 = 2x4 + 7x2 – 4 4x3 – 4x3 = 0 5y2 – 8y5 + 8y5 = 5y2 ¾x3 + 4x2 – x3 + 7 = -¼x3 + 4x2 + 7 -3p7 – 5p7 – p7 = -9p7 5.1

Missing Terms x3 – 5 is missing terms of x2 and x So what! Leaving space for missing terms will help you when you start adding & subtracting polynomials Write the expression above in either of two ways: With 0 coefficients: x3 + 0x2 + 0x – 5 With space left: x3 – 5 5.1

Adding 2 Polynomials - Horizontal To add polynomials, remove parentheses and combine like terms. (2x – 5) + (7x + 2) = 2x – 5 + 7x + 2 = 9x – 3 (5x2 – 3x + 2) + (-x – 6) = 5x2 – 3x + 2 – x – 6 = 5x2 – 4x – 4 This is called the horizontal method because you work left to right on the same “line” 5.1

Adding 2 Polynomials - Vertical To add polynomials vertically, remove parentheses, put one over the other lining up like terms, add terms. (2x – 5) + (7x + 2) = 2x – 5 + 7x + 2 Add the matching columns 9x – 3 (5x2 – 3x + 2) + (-x – 6) = 5x2 – 3x + 2 + – x – 6 5x2 – 4x – 4 This is called the vertical method because you work from top to bottom. More than 2 polynomials can be added at the same time. 5.1

Opposites of Polynomials The opposite of a polynomial has a reversed sign for each monomial The opposite of y + 36 is -y – 36 The opposite of -4x2 + 2x – 4 is 4x2 – 2x + 4 Polynomial: Opposite: -x + 2 x – 2 3z – 5y -3z + 5y ¾y5 + y5 – ¼y5 -¾y5 – y5 + ¼y5 -(x3 – 5) x3 – 5 5.1

Subtracting Polynomials To subtract polynomials, add the opposite of the second polynomial. (7x3 + 2x + 4) – (5x3 – 4) add the opposite! (7x3 + 2x + 4) + (-5x3 + 4) Use either horizontal or vertical addition. Sometimes the problem is posed as subtraction: x2 + 5x +6 make it addition x2 + 5x +6 - (x2 + 2x) _ of the opposite -x2 – 2x__ 3x +6 5.1

Subtractions for You 5.1

What Next? Section 5.2 – Multiplication of Polynomials 5.1