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Sect. 5.1 Polynomials and Polynomial Functions

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. Definitions.

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Sect. 5.1 Polynomials and Polynomial Functions

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  1. 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

  2. Definitions • An algebraictermis 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 polynomialexpression 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

  3. Ordering a Polynomial’s TermsIf 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

  4. 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 degreeis the same as the degree of the term in the polynomial with largestdegree (leading term?). 5.1

  5. 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

  6. Opposites of Monomials The opposite of a monomial has a different sign The opposite of 36 is -36 The opposite of -4x2is 4x2 Monomial:Opposite: -2 2 5y -5y ¾y5-¾y5 -x3x3 0 0 5.1

  7. 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

  8. 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

  9. 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 collectinglike terms, summing their coefficients • Let’s try: 6x2 – 2x2 – 3 + x2 – 11 • Sum of: 6x2 + -2x2 + x2 is5x2 • Sum of: -3 + -11 is-14 • Simplified polynomial is: 5x2 – 14 5.1

  10. 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

  11. 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

  12. 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

  13. 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 + 2Add the matching columns 9x – 3 • (5x2 – 3x + 2) + (-x – 6) = 5x2 – 3x + 2 + – x – 65x2 – 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

  14. 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

  15. 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

  16. Subtractions for You 5.1

  17. What Next? • Section 5.2 –Multiplication of Polynomials 5.1

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