Naming Polynomials Add and Subtract Polynomials Multiply Polynomials U2 – 2.1 Polynomials Naming Polynomials Add and Subtract Polynomials Multiply Polynomials

Definitions Exponents Power Simplify Terms Monomials Polynomials Exponents: the power attached to the base Power: the number that is the exponent written in the form xª Simplify: rewrite expression without parenthesis or negative exponents Terms: Each number or variable/number or variable in an expression Monomials: some algebraic expressions with exponents. Polynomials: Monomial or group of monomials.

More Definitions Like Terms Constant Degree Coefficient Binomial Trinomial Like Terms: two monomials that are the same or differ only by their coefficient. Constant: monomials that contain no variable Degree: sum of exponents on variables of one monomial Coefficient: Number in front of the variables Binomial: expression with 2 unlike terms Trinomial: expression with 3 unlike terms

a. 2 𝒑 𝟒 + 5 𝒑 𝟑 EXAMPLE 1 Identify polynomial functions Name each polynomial by degree (highest exponent), the number of terms (and expression that can be written as a sum, the parts added together) leading coefficient (number in front of the variable with highest degree) a. 2 𝒑 𝟒 + 5 𝒑 𝟑 SOLUTION a. The degree is 4, number of terms is 2, leading coefficient 2

a. 10a EXAMPLE 1 Identify polynomial functions Name each polynomial by degree, the number of terms, and leading coefficient. a. 10a SOLUTION a. The degree is 1, the number of terms is 1, and the leading coefficient is 10.

a. - 5 𝒏 𝟑 +10n - 2 EXAMPLE 1 Identify polynomial functions Name each polynomial by degree, the number of terms, leading coefficient a. - 5 𝒏 𝟑 +10n - 2 SOLUTION a. The degree is 5, the number of terms is 3, and the leading coefficient is 3.

a. 3 EXAMPLE 1 Identify polynomial functions Name each polynomial by degree, the number of terms, and leading coeffcient. a. 3 SOLUTION a. The degree is :None and the number of terms is 1, leading coefficient none.

1. f (x) = 13 – 2x GUIDED PRACTICE for Examples 1 and 2 State the polynomials degree, terms, and leading coefficient. 1. f (x) = 13 – 2x SOLUTION It is a polynomial function. Standard form: – 2x + 13 Degree: 1 Leading coefficient of – 2. Number of terms : 2 f (x) = – 2x + 13

2. p (x) = 9x4 – 5x 2 + 4 GUIDED PRACTICE for Examples 1 and 2 It is a polynomial function. Standard form: . p (x) = 9x4 – 5x 2 + 4 Degree: 4 Leading coefficient of 9. Number of terms : 3 SOLUTION

3. h (x) = 6x2 + π – 3x GUIDED PRACTICE for Examples 1 and 2 SOLUTION The function is a polynomial function that is already written in standard form will be 6x2– 3x + π . It has degree 2 and a leading coefficient of 6. It is a polynomial function. Standard form: 6 𝑥 2 – 3x + π Degree: 2 Terms: 3 Leading coefficient of 6

x3 + 6x2 + 11 in a vertical format. EXAMPLE 1 Add polynomials vertically and horizontally Add 2x3 – 5x2 + 3x – 9 and x3 + 6x2 + 11 in a vertical format. SOLUTION a. 2x3 – 5x2 + 3x – 9 + x3 + 6x2 + 11 3x3 + x2 + 3x + 2

b. Add 3y3 – 2y2 – 7y and – 4y2 + 2y – 5 in a horizontal format. EXAMPLE 1 Add polynomials vertically and horizontally b. Add 3y3 – 2y2 – 7y and – 4y2 + 2y – 5 in a horizontal format. (3y3 – 2y2 – 7y) + (– 4y2 + 2y – 5) = 3y3 – 2y2 – 4y2 – 7y + 2y – 5 = 3y3 – 6y2 – 5y – 5

EXAMPLE 2 Subtract polynomials vertically and horizontally a. Subtract 3x3 + 2x2 – x + 7 from 8x3 – x2 – 5x + 1 in a vertical format. 8x3 – x2 – 5x + 1 – (3x3 + 2x2 – x + 7) 8x3 – x2 – 5x + 1 + – 3x3 – 2x2 + x – 7 5x3 – 3x2 – 4x – 6 SOLUTION a. Align like terms, then add the opposite of the subtracted polynomial.

4z2 + 9z – 12 in a horizontal format. EXAMPLE 2 Subtract polynomials vertically and horizontally Subtract 5z2 – z + 3 from 4z2 + 9z – 12 in a horizontal format. (4z2 + 9z – 12) – (5z2 – z + 3) = 4z2 + 9z – 12 – 5z2 + z – 3 = 4z2 – 5z2 + 9z + z – 12 – 3 = – z2 + 10z – 15 Write the opposite of the subtracted polynomial, then add like terms.

1. (t2 – 6t + 2) + (5t2 – t – 8) GUIDED PRACTICE for Examples 1 and 2 Find the sum or difference. 1. (t2 – 6t + 2) + (5t2 – t – 8) t2 – 6t + 2 + 5t2 – t – 8 SOLUTION 6t2 – 7t – 6

2. (8d – 3 + 9d3) – (d3 – 13d2 – 4) GUIDED PRACTICE for Examples 1 and 2 2. (8d – 3 + 9d3) – (d3 – 13d2 – 4) = (8d – 3 + 9d3) – (d3 – 13d2 – 4) = (8d – 3 + 9d3) – d3 + 13d2 + 4) SOLUTION = 9d3 –3 d3 + 13d2 + 8d – 3 + 4 = 8d3 + 13d2 + 8d + 1

( 3 𝑋 2 - 5 ) + ( 7 𝑋 2 - 3 ) ( 5 𝐶 2 + 7c + 3 ) + ( 𝑐 2 + 6c + 4 ) TRY THE FOLLOWING PROBLEMS ( 3 𝑋 2 - 5 ) + ( 7 𝑋 2 - 3 ) ( 5 𝐶 2 + 7c + 3 ) + ( 𝑐 2 + 6c + 4 ) ( − 𝑥 4 +5 𝑥 5 - 3 𝑥 3 - 2x + 2) + ( 𝑥 3 + 𝑥 3 + 6 𝑥 5 + 4 )

ANSWERS TO ADDITION PROBLEMS ( 3 𝑋 2 - 5 ) + ( 7 𝑋 2 - 3 ) = 10 𝑋 2 - 8 ( 5 𝐶 2 + 7c + 3 ) + ( 𝑐 2 + 6c + 4 ) = 6 𝑥 2 + 13c + 7 ( − 𝑥 4 +5 𝑥 5 - 3 𝑥 3 - 2x + 2) + ( 𝑥 3 + 𝑥 3 + 6 𝑥 5 + 4 ) = 11 𝑥 5 −𝑥 4 −𝑥 3 -2x + 6

( 4 𝑋 2 - 9 ) - ( 2 𝑋 2 - 3 ) ( 4 𝐶 2 + 2c + 6 ) - ( 𝑐 2 + 8c + 4 ) TRY THE FOLLOWING PROBLEMS ( 4 𝑋 2 - 9 ) - ( 2 𝑋 2 - 3 ) ( 4 𝐶 2 + 2c + 6 ) - ( 𝑐 2 + 8c + 4 ) ( − 2𝑥 4 +2 𝑥 5 - 8 𝑥 3 - 3x + 9 ) - ( - 𝑥 3 + 𝑥 3 + 4 𝑥 5 + 5 )

ANSWERS TO THE SUBTRACT PROBLEMS ( 4 𝑋 2 - 9 ) - ( 2 𝑋 2 - 3 ) = 2𝑥 2 - 6 ( 4 𝐶 2 + 2c + 6 ) - ( 𝑐 2 + 8c + 4 ) = 3 𝑐 2 +6𝑐+2 ( − 2𝑥 4 +2 𝑥 5 - 8 𝑥 3 - 3x + 9 ) - ( - 𝑥 3 + 𝑥 3 + 4 𝑥 5 + 5 ) = −2𝑥 5 −2𝑥 4 −8𝑥 3 -3x + 4

Multiply Polynomials 3(2) 2.3(2x) 3.3x(2x) 4.3x(2x+1)