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Rewrite a polynomial

EXAMPLE 1. Rewrite a polynomial. Write 15 x – x 3 + 3 so that the exponents decrease from left to right. Identify the degree and leading coefficient of the polynomial. SOLUTION. Consider the degree of each of the polynomial’s terms. 15 x – x 3 + 3.

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Rewrite a polynomial

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  1. EXAMPLE 1 Rewrite a polynomial Write 15x – x3 + 3 so that the exponents decrease from left to right. Identify the degree and leading coefficient of the polynomial. SOLUTION Consider the degree of each of the polynomial’s terms. 15x – x3 + 3 The polynomial can be written as – x3 +15 + 3. The greatest degree is 3, so the degree of the polynomial is 3, and the leading coefficient is –1.

  2. Expression Is it a polynomial? Classify by degree and number of terms a. 9 Yes 0 degree monomial b. 2x2 + x – 5 Yes 2nd degree trinomial c. 6n4 – 8n No; variable exponent d. n– 2 – 3 No; variable exponent e. 7bc3 + 4b4c Yes 5th degree binomial EXAMPLE 2 Identify and classify polynomials Tell whetheris a polynomial. If it is a polynomial, find its degree and classify it by the number of its terms. Otherwise, tell why it is not a polynomial.

  3. EXAMPLE 3 Add polynomials Find the sum. a. (2x3 – 5x2 + x) + (2x2 + x3 – 1) b. (3x2 + x – 6) + (x2 + 4x + 10)

  4. + x3 + 2x2 – 1 EXAMPLE 3 Add polynomials SOLUTION a. Vertical format: Align like terms in vertical columns. (2x3 – 5x2 + x) 3x3 – 3x2 + x – 1 b. Horizontal format: Group like terms and simplify. (3x2 + x – 6) + (x2 + 4x + 10) = (3x2+ x2) + (x+ 4x) + (– 6+ 10) = 4x2 + 5x + 4

  5. 1. Write 5y – 2y2 + 9 so that the exponents decrease from left to right. Identify the degree and leading coefficient of the polynomial. 2. Tell whether y3 – 4y + 3 is a polynomial. If it is a polynomial, find its degree and classify it by the number of its terms. Otherwise, tell why it is not a polynomial. ANSWER – 2y2 +5y + 9 Degree: 2, Leading Coefficient: –2 ANSWER polynomial Degree: 3, trinomial EXAMPLE 1 for Examples 1,2, and 3 Rewrite a polynomial GUIDED PRACTICE

  6. 3. Find the sum. ANSWER = 8x3 + 4x2+ 2x – 6 EXAMPLE 3 for Example for Examples 1,2, and 3 Add polynomials GUIDED PRACTICE (5x3 + 4x – 2x) + (4x2 +3x3 – 6)

  7. EXAMPLE 4 Subtract polynomials Find the difference. a. (4n2 + 5) – (–2n2 + 2n – 4) b. (4x2 – 3x + 5) – (3x2 – x – 8)

  8. –(–2n2 + 2n – 4) 2n2 – 2n + 4 EXAMPLE 4 Subtract polynomials SOLUTION a. (4n2 + 5) 4n2 + 5 6n2 – 2n + 9 b. (4x2 – 3x + 5) – (3x2 – x – 8) = 4x2 – 3x + 5– 3x2 + x + 8 = (4x2– 3x2) +(–3x+x) + (5+ 8) =x2–2x+13

  9. EXAMPLE 5 Solve a multi-step problem BASEBALL ATTENDANCE Major League Baseball teams are divided into two leagues. During the period 1995–2001, the attendance Nand A (in thousands) at National and American League baseball games, respectively, can be modeled by N = –488t2 + 5430t + 24,700 and A = –318t2 + 3040t + 25,600 where tis the number of years since 1995. About how many people attended Major League Baseball games in 2001?

  10. EXAMPLE 5 Solve a multi-step problem SOLUTION STEP 1 Add the models for the attendance in each league to find a model for M, the total attendance (in thousands). M =(–488t2 + 5430t + 24,700) +(–318t2 + 3040t + 25,600) = (–488t2– 318t2) + (5430t+ 3040t) + (24,700 + 25,600) = –806t2 + 8470t + 50,300

  11. M = –806(6)2 + 8470(6) + 50,300 72,100 ANSWER About 72,100,000 people attended Major League Baseball games in 2001. EXAMPLE 5 Solve a multi-step problem STEP 2 Substitute 6 for tin the model, because 2001 is 6 years after 1995.

  12. 4. Find the difference. BASEBALL ATTENDNCE Look back at Example 5. Find the difference in attendance at National and American League baseball games in 2001. 5. ANSWER –x2 – 11x + 9 ANSWER about 7,320,000 people EXAMPLE 4 for Examples 4 and 5 Subtract polynomials GUIDED PRACTICE a. (4x2 – 7x) – (5x2 + 4x – 9)

  13. No; one exponent is not a whole number. ANSWER ANSWER 8th degree trinomial Daily Homework Quiz If the expression is a polynomial, find its degree and classify it by the number of terms. Otherwise, tell why it is not a polynomial. 1. m3 + n4m2 + m–2 2. – 3b3c4 – 4b2c + c8

  14. ANSWER 4m2 + 5 ANSWER –5a2 + a + 5 Daily Homework Quiz Find the sum or difference. 3. (3m2 – 2m + 9) + (m2 + 2m– 4) 4. (– 4a2 + 3a – 1) – (a2 + 2a – 6)

  15. ANSWER about 185 dogs and cats Daily Homework Quiz 5. The number of dog adoptions D and cat adoptions C can be modeled by D = 1.35 t2 – 9.8t + 131 and C= 0.1t2 – 3t + 79 where t represents the years since 1998. About how many dogs and cats were adopted in 2004?

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